Mathematics

"In 1920 logic was mostly a philosopher's garden. There were also a few mathematicians there, cultivating the logical roots of the mathematical tree. Today, Recursion Theory, Set Theory, Model Theory and Proof Theory, logic's major subdisciplines, have become full-fledged branches of mathematics."

"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."

"Modern descriptive set theory builds on both the classical descriptive set theory and on recursion theory. It has become clear in the 1950’s that the topological approach of classical descriptive set theory and the recursion theoretic techniques of logical definability describe the same phenomena. Modern descriptive set theory unified both approaches, as well as the notation."

"Does set theory, once we get beyond the integers, refer to an existing reality, or must it be regarded, as formalists would regard it, as an interesting formal game? ... A typical argument for the objective reality of set theory is that it is obtained by extrapolation from our intuitions of finite objects, and people see no reason why this has less validity. Moreover, set theory has been studied for a long time with no hint of a contradiction. It is suggested that this cannot be an accident, and thus set theory reflects an existing reality. In particular, the Continuum Hypothesis and related statements are true or false, and our task is to resolve them."

"Frege's development began with... "an epoch-making little book" called ', translated as Concept Script or Conceptual Notation... Just as Aristotle's ' is the foundation of traditional or syllogistic logic—the logic of the categorical three-term syllogism, Frege's Begriffsschrift is the keystone of modern or mathematical logic."

"If 99.99% of the universe is in the plasma state then for the remaining 0.01% must we learn a whole discipline of Fluid Mechanics or Hydrodynamics? The answer is No! More often than not, hydrodynamics provides a first level description of an astrophysical fluid."

"Fluid mechanics is a part of applied mathematics, of physics, of many branches of engineering, certainly civil, mechanical, chemical, and aeronautical engineering, and of naval architecture and geophysics, with astrophysics and biological and physiological fluid dynamics to be added."

"Curiosity about at least two of the branches of fluid mechanics and their applications has a long and distinguished history, for in the Proverbs of Solomon the son of David, king of Israel, it was stated in the words of Agur the son of Jakeh that "There be three things which are too wonderful for me, yea four which I know not," of which two were "The way of an eagle in the air" and "The way of a ship in the midst of the sea," which I take to be questions of aerodynamics and naval architecture, questions that concern us still."

"One wonders if the foundations of fluid mechanics or gas dynamics would ever have been secured if Euler or Boltzmann had justified their work to themselves in terms of its contribution to the design of jet aircraft."

"The arch is one of those brilliant innovations... Spanning... with horizontal beams is a losing game. ...By converting all the stress that fractures the middle of... stone beams—technically tension—into compression on stone piers larger... spaces could be spanned. ...But shift the pressure even slightly off center, and the pillar is likely to collapse. ...In their early incarnations, the limitations of both arch and dome was the ability of craftsmen to shape the stones carefully enough to create blocks precisely in the wedge shapes needed for a particular arch. Despite their mathematical sophistication in most other respects, the architects of antiquity lacked a proper geometric solution to the ideal form of the arch. (It was not until 1675 that the English polymath Robert Hooke described mathematically the shape of an arch loaded in pure compression, that is, with no tension, by showing how it describes an upside-down version of the catenary curve of a hanging chain.) As a result, the only way they could design an arch, and its component stones, was completely by eye, and... such tolerances commanded high prices. Rome overcame this drawback with typical ingenuity, first replacing stones and mortar... and expensive stonecutters with relatively cheap bricklayers. Even more ingeniously, some anonymous Roman builder found how to combine the mortar—in Latin pulvis puteoli—with lime, sand, and gravel to make the first concrete. ...The concrete domes of Rome were not surpassed until the age of steel."

"Certainly the most striking contemporary example of a similar form [ arch] is to be found in St. Louis' ... In its incredible scale and construction out of metal plates this structure also serves as a convenient reminder of the important developments of the production of iron and steel that took place during the Industrial Revolution and that have so significantly affected arches as well as all other types of structural forms for the past 150 years."

"I love the catenary because it tells the story of holding up the roof."

"The most difficult properties of the Catenary were revealed before the close of the seventeenth century. This curve is entitled to particular attention, not only because it throws light on the theory of arches, but because it applies directly to the construction of suspended bridges, which are now deservedly coming into repute."

"The Catenary has its name from Catena, a chain; being the curve which a regular and very flexible chain will assume, if suspended loosely from both ends. It seems to have been first noticed by the famous Galileo, who proposed it as the figure of an arch of equilibration, but unfortunately mistook it for a . In fact, the Catenary, near its vertex, differs insensibly from that curve, but afterwards deviates more considerably. ...the Parabola diverges faster from its axis than the Catenary. The error of Galileo in confounding those two curves was not perceived till , in 1669, ascertained, by actual experiment, that the Catenary is neither a Parabola nor an . It was in 1691, that the penetrating genius of James Bernoulli discovered the true nature of the catenarian curve. A similar investigation was soon produced by John Bernoulli, by Huygens, and by Leibnitz. This latter philosopher, whose powers of invention and stores of learning were alike transcendant, discovered the fine relation of the Catenary to the Logarithmic Curve."

"A non-catenary curve might be perfectly doable, but it takes more material, it has bigger beam sections, and overall it is much more complicated to construct... Even if the cladding falls out, the interiors and everything else falls away and the whole thing turns to dust and rubble and sand, [the catenary] should still stand."

"Concrete being such a fluid and dynamic material... finds its identity once it is contained. ...A few... who used the forming materials at hand [were]... Antoni Gaudi... ... ... Felix Candela... ... ... Miguel Fisac... Many of these early innovators pushed the computational envelope... Some, like Antoni Gaudi, looked to nature for inspiration. The question... Do we need to "reinvent forming" or just draw from nature, i.e., gravity—catenary action? as Gaudi did. Alan Chandler in fabric framework notes "...for Felix Candela and Christopher Alexander fabric acted as a permanent shutter (framework)..." Chandler speaks of the family of fabric construction that includes... s... Pneumatic structures... Hydrostatic structures and... Shell structures derived from membrane form-finding. When faced with extremely complicated and complex shapes Heinz Isler and Antoni Gaudi used fabric as a modeling tool. These visionaries recognized that hanging chains and fabrics, forming catenaries, are in pure tension and when inverted are in pure compression and very stable. Gaudi, whose ing preceded the works of Candela... looked to nature and natural forms—an approach today called biomimicry..."

"What has been objected by an anonymous author, in the Leipsic Acts of Feb. 1699, in his animadversions on my demonstrations concerning the catenary, is this: that I have undertaken to demonstrate, after my manner, a matter found out and published by others seven years ago. This is true, and I cannot find any thing in it that is blame worthy. Those great men Huygens, Leibnitz, and Bernouilli, have discovered and communicated many properties of the catenaria, but without demonstration. I have contrived demonstrations, which was the thing I undertook to do. But was this matter that is the nature and primary properties of the catenaria all found out and published by others? ...From all ages architects have made use of arches in public buildings, as well for strength as beauty. Yet what was the true geometrical figure of an arch was not known before my demonstrations came out."

"It was comparatively late that the theory of arches attracted the notice of mathematicians. Dr. Hooke gave the hint, that the figure of a perfectly flexible cord or chain, suspended from two points, was the proper form for an arch. Galileo considered the catenary as a parabolic curve, and John Bernouilli appears to have been the first who discovered its nature. Dr. Gregory (Phil. Trans. 1697) published an investigation of its properties, and observes that the inverted catenary is the best form for an arch on account of its lightness. This is true so long as it is not pressed by an extraneous weight. It is not, however, capable of bearing a load on any part, much less of being filled up on the spandrels, which must be the case in practice. Other considerations must be involved before it can be fitted to receive a roadway or other weight, either upon its crown or haunches."

"Corol. 6.—In a vertical plane, but in an inverted situation, the chain will preserve its figure without falling, and therefore will constitute a very thin arch or fornix: that is, infinitely small, rigid, polished spheres, disposed in an inverted curve of a catenaria, will form an arch no part of which will be thrust outwards or inwards by other parts, but, the lowest parts remaining firm, it will support itself by means of its figure... none but the catenaria is the figure of the true and legitimate arch or fornix. And when the arches of other figures is supported, it is because in their thickness some catenaria is included. ...From Corol. 5... it may be collected, by what force an arch or buttress presses a wall outwardly, to which it is applied. For this is the same with that part of the force sustaining the chain, which draws according to a horizontal direction. For the force which in the chain draws inwards, in an arch equal to the chain drives outwards. All other circumstances, concerning the strength of walls to which arches are applied, may be geometrically determined from this theory, which are the chief things in the construction of edifices."

"The trolley-wire must... be suspended with only a very small sag, and to obtain this result without excessive tension in the wire the span must be relatively short, i.e. of the order of 10 ft. to 15 ft. It is obvious that for these short spans the method of construction adopted in tramway practice would be unsuitable, both from the mechanical as well as the electrical standpoint. However, by adopting the catenary system—that is, supporting the trolley-wire from another wire, suspended with considerable sag between supports of moderate span—we are able to obtain a level trolley-wire with a relatively small number of supporting structures. The wire from which the trolley-wire is supported is called the "catenary" or "messenger" wire, and by insulating this wire from the supporting structures there is no necessity for insulated hangers on the trolley-wire."

"One can roll noncircular wheels over appropriate road surfaces. The most striking example of this is the fact that a can roll on a road that consists of linked catenaries (the catenaries are defined by y = - cosh x) ...(the ride is not smooth in the sense that the center does not move forward at a constant rate of rotation, but... the actual ride... feels quite smooth). This animation was inspired by an exhibit at San Francisco's ..."

"Within a shed erected on the construction site of the church of the Sagrada family... Gaudí... made an upside-down model using lightweight cables to represent the structural lines of the future church—a model based on the structural notion of the inverted catenary. ...Analogically represented by little pouches filled with lead pellets the action of the stresses has been done ...The resulting chain configurations are used to determine the geometrical shapes and structural profiles of columns, pillars, arches, and vaults. ...Vicens Vilarrubias i Valls took photos of the model ...Gaudí used these photos upside-down to draw over them the external and internal elevations, studies of details and sections of the building."

"The flexible chain, hanging under the action of applied force, will assume a certain shape, namely the catenary if the chain is subjected only to its own weight, or a if the load is uniformly distributed horizontally. Whatever the load, there will be a corresponding shape, and the structural action in all cases is the same; purely tensile forces are transmitted along the centre line of the chain."

"As Hooke saw in 1675 with his ut pendet continuum flexile, sic stabit contiguum rigidum inversum, ...a hanging chain may be inverted to give a satisfactory arch to carry the same loads, but working in compression rather than tension. The compressive arch, however, if of vanishingly small thickness, would be in unstable equilibrium, and stability is conferred in practice by making the arch ring of finite depth. Now if purely compressive forces, without bending, are to be transmitted from one portion of the arch to the next (as purely tensile forces are transmitted in the chain), then the arch centre line can accept only a single type of loading. Thus a parabolic arch can carry only a uniformly distributed horizontal load (although the magnitude of the load is arbitrary). It is the depth in a real arch which enables the arch to carry wider ranges of loading; a large number of different idealized centre-line arches can be contained within a given practical profile. ...[T]his must be so, or no mediaeval bridge would have survived its decentering."

"Gaudí was drawn not just to the aesthetics of the catenary but also to what it represented mathematically. His use of catenaries made the structural mechanics of a building a principal feature of its design. Gaudí realized that the entire architecture of a building could be drafted using a model of hanging chains... when he was commissioned to design a church the Colònia Güell... he made an upside-down skeleton of the project. Instead of using metal chains, he used string weighed down by hundreds of sachets containing lead shot. The weight of each sachet on the string created a mesh of 'transformed' catenary curves. The arches of these transformed catenaries were the most stable curves to withstand a corresponding weight at the same position (such as the roof, or building materials)."

"In 1690... Jacob Bernoulli brought up the problem of the catenary in a memoir... in the '...Huygens' solution represents the past... a complex, though skillful, geometrical method. Leibniz, using his new [infinitesimal calculus] reaches a correct analytical formula...y/a = (b^\frac{x}{a} + b^\frac{-x}{a})/2 where a is [a] segment... and b... corresponds to... e... ...supplied two correct constructions ...presents valid statistical arguments and... new and important... equations of equilibrium in differential form. ...In 1697-1698, Jacob Bernoulli was the first to derive the general equations that not only solved the problem, but also permitted the treatment of the more general theme of the equilibrium of a flexible rope, subject to any distribution of tangential (f_t) and normal (f_n) forces. Bernoulli's equations are...\frac{dT}{ds} + f_t = 0, \qquad \frac{T}{r} + f_n= 0where T is the tension, s the curvilinear abscissa, and r the radius of curvature."

"The true Mathematical and Mechanical Form of all manner of Arches for building with the true butment necessary to each of them, a Problem which no Architectonick Writer hath ever yet attempted, much less perform'd. ...Ut pendet continaum flexile, sic stabit contiguum rigidum, which is the Linea Catenaria."

"The largest catenary structure of masonry is the Great Hall of the Palace of Taq Kisra, at Qesiphon, then the capital of Persia."

"I will begin with the subject of your bridge... and it is with great pleasure that I learn... that the execution of the arch of experiment exceeds your expectations. ...You hesitate between the catenary and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium. ...I would propose that you make your middle rail an exact catenary, and the interior and exterior rails parallels to that. It is true, they will not be exact catenaries, but they will depart very little from it; much less than portions of circles will."

"Homological algebra, conceived as a general tool reaching beyond all special cases, was invented by Cartan and Eilenberg (their book “Homological Algebra” appeared in 1956). This book is a very precise exposition, but limited to the theory of modules over rings and the associated functors “Ext” and “Tor”."

"Modular forms have long played a key role in the theory of numbers, including most famously the proof of Fermat's Last Theorem."

"Branching processes provide perhaps the simplest example of a phase transition. They occur naturally as a model of the random evolution of a population that changes in time as a result of births and deaths."

"[A]s a liquid changes into a gas at the critical temperature T_c, the heat capacity diverges as c \sim \frac{1}{\left|T-T_c\right|^{0.11008 \ldots}}. The exponent is not known precisely. It is thought not to be a rational number, but should instead be viewed as a universal mathematical constant, similar to \pi"

"The end of the fifties marked somewhat of a watershed for continuous time Markov chains, with two branches emerging a theoretical school following Doob and Chung, attacking the problems of continuous-time chains through their sample paths, and using measure theory, martingales, and stopping times as their main tools; and an applications oriented school following Kendall, Reuter and Karlin, studying continuous chains through the transition function, enriching the field over the past thirty years with concepts such as reversibility, ergodicity, and stochastic monotonicity inspired by real applications of continuous-time chains to queueing theory, demography, and epidemiology."

"The mathematical theory of critical phenomena is currently undergoing intense development. Intertwined with the science of phase transitions, it draws on ideas from probability theory and statistical physics."

"Combinatorics and Algebraic Geometry have enjoyed a fruitful interplay since the nineteenth century. Classical interactions include invariant theory, theta functions, and enumerative geometry."

"The result of a counting process is independent of the way in which it is done. This is really an assumption about what counting is. It would be senseless to try to prove it, because it is so basic; either you see it or you don't—but in the latter case, a proof won't help you a bit."

"Ergodic theory is a mathematical subject that studies the statistical properties of deterministic dynamical systems. It is a combination of several branches of pure mathematics, such as measure theory, functional analysis, topology, and geometry, and it also has applications in a variety of fields in science and engineering, as a branch of applied mathematics."

"The study of waves is important to virtually every branch of science and engineering. Indeed, waves are also important to everyday life. Sound waves allow us to hear, and electromagnetic waves allow us to see."

"The mass gap is the reason, if you will, that we do not see classical nonlinear Yang-Mills waves. They are a good approximation only under inaccessible conditions. I have spent most of my career wishing that we had a really good way to quantitatively understand the mass gap in four-dimensional gauge theory. I hope that this problem will be solved one day."

"One morning early in my (Hersh's) years as a thesis student of Peter Lax, I entered my mentor's office to find him glowing in smiles. “Louis is back!” he cried out to me. Louis, I wondered? Oh yes, Louis Nirenberg, also one of the partial differential specialists on the faculty of NYU's Courant Institute. He had been on leave in England; now he was back home! At the time. I didn't get it. Louis Nirenberg and Peter Lax were grad students together at NYU. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations, and Peter, a world master of hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success."

"Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods."

"The first systematic attack on a problem involving a partial differential equation was carried out in a sequence of 1746 papers by Jean Le Rond d'Alembert (1717-1783), who sought the fundamental modes of vibration of a vibrating string."

"The development of the theory of P.D.E. is closely linked with advances in complex analysis; in fact, Riemann’s approach to the study of conformal mapping via the Dirichlet principle led to the systematic development of the theory of elliptic P.D.E. and associated variational problems. The application of these methods to the theory of several complex variables was initiated by Hodge in his theory of harmonic integrals on compact manifolds. It is this work that led H. Weyl to prove the fundamental hypoellipticity theorem, known as Weyl’s lemma, which in turn led to the development of the general theory of elliptic P.D.E."

"If the original work of Calderón and Zygmund was related to elliptic PDE's, later developments allowed applications of their theory to parabolic equations and to general hypoelliptic operators, and the more recent explosion of interest in the theory of oscillatory integrals and in problems involving curvature has much to do with hyperbolic equations."

"CFD merely provides tools for solving the equations of fluid motion; it does not change the conceptual landscape in any fundamental way. Still, it is so powerful that it has become indispensable to the practice of aeronautical engineering."

"Computational fluid dynamics (CFD) techniques are used to simulate a number of phenomena. Obviously, crashing waves and oceans can leverage CFD, but explosions, fireballs, and smoke effects all make use of CFD nowadays as well."

"Automorphic forms can be thought of as fundamental particles in harmonic analysis, which deals in part with waves and frequencies. In a symphony orchestra, automorphic forms issue instructions and work with eigenvalues — the different speeds a violin string moves when struck, for instance — to produce the notes played."

"He liked putting different pieces of mathematics together: geometry, analysis, topology… so automorphic forms should have appealed to him. But for some reason he didn’t get interested in that at the time. I think the junction between Grothendieck and Langlands was realized only in 1972 at Antwerp. Serre had given a course on Weil’s theorem in 1967–1968. But after 1968 Grothendieck had other interests. And before 1967 things were not ripe. I’m not sure."

"At the heart of Langlands' program is the general notion of an "automorphic representation" π and its L-function L(s, π ) ... both defined via group theory and the theory of harmonic analysis on so-called adele groups ... The conjectures of Langlands ... amount (roughly) to the assertion that the other zeta-functions arising in number theory are but special realizations of these L(s, π ). Herein lies the agony as well as the ecstasy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and p-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding."