Differential Geometry Quotes

the seven “liberal arts”: Grammar, the foundation of science; Logic, which differentiates the true from the false; Rhetoric, the source of law; Arithmetic, the foundation of order because “without numbers there is nothing”; Geometry, the science of measurement; Astronomy, the most noble of the sciences because it is connected with Divinity and Theology; and lastly Music.

Even there, something inside me (and, I suspect, inside many other computer scientists!) is suspicious of those parts of mathematics that bear the obvious imprint of physics, such as partial differential equations, differential geometry, Lie groups, or anything else that's “too continuous.

In physics, theories are made of math. We don’t use math because we want to scare away those not familiar with differential geometry and graded Lie algebras; we use it because we are fools. Math keeps us honest—it prevents us from lying to ourselves and to each other. You can be wrong with math, but you can’t lie.

Is it possible that mathematical pathology, i.e. chaos, is health? And that mathematical health, which is the predictability and differentiability of this kind of a structure, is disease?

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

In the 1950s, John Nash disrupted the balance between geometry and analysis when he discovered that the abstract geometric problem of isometric embedding could be solved by the fine “peeling” of partial differential equations.

At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, “out of a curiosity to see what there was in it.” He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply.

In the intricate and mutable space-time geometry at the black hole, in-falling matter and energy interacted with the virtualities of the vacuum in ways unknown to the flatter cosmos beyond it. Quasi-stable quantum states appeared, linked according to Schrodinger's wave functions and their own entanglement, more and more of them, intricacy compounding until it amounted to a set of codes. The uncertainty principle wrought mutations; variants perished or flourished; forms competed, cooperated, merged, divided, interacted; the patterns multiplied and diversified; at last, along one fork on a branch of the life tree, thought budded. That life was not organic, animal and vegetable and lesser kingdoms, growing, breathing, drinking, eating, breeding, hunting, hiding; it kindled no fires and wielded no tools; from the beginning, it was a kind of oneness. An original unity differentiated itself into countless avatars, like waves on a sea. They arose and lived individually, coalesced when they chose by twos or threes or multitudes, reemerged as other than they had been, gave themselves and their experiences back to the underlying whole. Evolution, history, lives eerily resembled memes in organic minds. Yet quantum life was not a series of shifting abstractions. Like the organic, it was in and of its environment. It acted to alter its quantum states and those around it: action that manifested itself as electronic, photonic, and nuclear events. Its domain was no more shadowy to it than ours is to us. It strove, it failed, it achieved. They were never sure aboardEnvoy whether they could suppose it loved, hated, yearned, mourned, rejoiced. The gap between was too wide for any language to bridge. Nevertheless they were convinced that it knew something they might as well call emotion, and that that included wondering.

pure mathematics, but these were very great indeed, and were indispensable to much of the work in the physical sciences. Napier published his invention of logarithms in 1614. Co-ordinate geometry resulted from the work of several seventeenth-century mathematicians, among whom the greatest contribution was made by Descartes. The differential and integral calculus was invented independently by Newton and Leibniz; it is the instrument for almost all higher mathematics. These are only the most outstanding achievements in pure mathematics; there were innumerable others of great importance.

Indeed, much of Newton’s intellectual development can be attributed to this tension between rationalism and mysticism. At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, ‘out of a curiosity to see what there was in it.’ He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

He has translated Virgil’s Aeneid . . . the whole of Sallust and Tacitus’ Agricola . . . a great part of Horace, some of Ovid, and some of Caesar’s Commentaries . . . besides Tully’s [Cicero’s] Orations. . . . In Greek his progress has not been equal; yet he has studied morsels of Aristotle’s Politics, in Plutarch’s Lives, and Lucian’s Dialogues, The Choice of Hercules in Xenophon, and lately he has gone through several books in Homer’s Iliad. In mathematics I hope he will pass muster. In the course of the last year . . . I have spent my evenings with him. We went with some accuracy through the geometry in the Preceptor, the eight books of Simpson’s Euclid in Latin. . . . We went through plane geometry . . . algebra, and the decimal fractions, arithmetical and geometrical proportions. . . . I then attempted a sublime flight and endeavored to give him some idea of the differential method of calculations . . . [and] Sir Isaac Newton; but alas, it is thirty years since I thought of mathematics.

The ramifications of Pythagoras' theorem have revolutionized twentieth century theoretical physics in many ways. For example, Minkowski discovered that Einstein's special theory of relativity could be represented by four-dimensional pseudo-Euclidean geometry where time is represented as the fourth dimension and a minus sign is introduced into Pythagoras' law. When gravitation is present, Einstein proposed that Minkowski's geometry must be "curved", the pseudo-Euclidean structure holding only locally at each point. A complex vector space having a natural generalization of the Pythagorean structure (defined over functions in an abstract space rather than geometrical points in the familiar Euclidean space) is known as Hilbert space and forms the basis of quantum mechanics. It is remarkable to think that the two pillars of twentieth century physics, relativity and quantum theory, both have their basis in mathematical structures based on a theorem formulated by an eccentric mathematician over two and a half thousand years ago.

Joining the world of shapes to the world of numbers in this way represented a break with the past. New geometries always begin when someone changes a fundamental rule. Suppose space can be curved instead of flat, a geometer says, and the result is a weird curved parody of Euclid that provides precisely the right framework for the general theory of relativity. Suppose space can have four dimensions, or five, or six. Suppose the number expressing dimension can be a fraction. Suppose shapes can be twisted, stretched, knotted. Or, now, suppose shapes are defined, not by solving an equation once, but by iterating it in a feedback loop. Julia, Fatou, Hubbard, Barnsley, Mandelbrot-these mathematicians changed the rules about how to make geometrical shapes. The Euclidean and Cartesian methods of turning equations into curves are familiar to anyone who has studied high school geometry or found a point on a map using two coordinates. Standard geometry takes an equation and asks for the set of numbers that satisfy it. The solutions to an equation like x^2 + y^2 = 1, then, form a shape, in this case a circle. Other simple equations produce other pictures, the ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number goes in, and so on, points hopping from place to place. A point is plotted not when it satisfies the equation but when it produces a certain kind of behavior. One behavior might be a steady state. Another might be a convergence to a periodic repetition of states. Another might be an out-of-control race to infinity.

Spengler's book is rich in these "morphological relationships" between dissimilar activities that prove the coherent spirit of each culture and epoch. So there was a common spirit int eh ancient Greek polis and in Euclidean geometry, as there was also between the differential calculus and the state of Louis XIV. Chronological "contemporaneity" was misleading. It should be replaced by an understanding of how different events play similar roles in expressing the culture spirit. Thus he sees his own kind of "contemporaneity" in the Trojan War and the Crusades, in Homer and the songs of the Nibelungs.

Who amongst them realizes that between the Differential Calculus and the dynastic principle of politics in the age of Louis XIV, between the Classical city-state and the Euclidean geometry, between the space perspective of Western oil painting and the conquest of space by railroad, telephone and long range weapon, between contrapuntal music and credit economics, there are deep uniformities?

We will focus attention on binary relations as these are by far the most important. If R ⊆ S × S is a binary relation on S, it is common to use the notation aRb in place of (a, b) ∈ R.

Furthermore, what you can say about what you saw depends on the structure of your symbolism — whether you describe it in English, Persian, Chinese, Euclidean geometry, non-Euclidean geometry, differential calculus or quaternions. This explains why, in Dr. Jones's words, "whatever we are describing, the human mind cannot be parted from it.

Most of his predecessors had considered the differential calculus as bound up with geometry, but Euler made the subject a formal theory of functions which had no need to revert to diagrams or geometrical conceptions.

All forces are multi-dimensional vibrations. The geometry and dynamics of dimensions will become important going forward. In a hologram, what is the location of the information? The holomovement is not static and the information is not perfectly distributed. Space and time are built up from entanglement. Consciousness is also entangled. The fractal nesting of event horizons For the photon, the dimensions involved are not the same as they are for the electron. The idea that all behaviors occupy the same spacetime is probably not correct. These forces and fields shape the spacetime they occupy in particular ways. In other words dimensions are dynamic and could also be virtual in some respect. There is a dynamic tension between space and time. Could the density of time create the pressure we call gravity? Does scale have a fractal quality? The fractal nesting of the relationship between space and time at different scales causes phase transitions as the influence of one force changes with respect to another. From this we get the astonishing variety of behaviors in the material world. I suspect that as the complexity increases, the dynamics of the dimensions is affected. The photon or electron is not a thing, it is a description of a relationship. The universe appears to be differentiating into a fractal computational geometry. Spacetime is fractal in the golden mean.