Coordinate Geometry Quotes

Is Euclidian geometry true or is Riemann geometry true? He answered, The question has no meaning. As well ask whether the metric system is true and the avoirdupois system is false; whether Cartesian coordinates are true and polar coordinates are false. One geometry can not be more true than another; it can only be more convenient. Geometry is not true, it is advantageous.

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.

Descartes was a philosopher, a mathematician, and a man of science. In philosophy and mathematics, his work was of supreme importance; in science, though creditable, it was not so good as that of some of his contemporaries. His great contribution to geometry was the invention of co-ordinate geometry, though not quite in its final form. He used the analytic method, which supposes a problem solved, and examines the consequences of the supposition; and he applied algebra to geometry. In both of these he had had predecessors—as regards the former, even among the ancients. What was original in him was the use of co-ordinates, i.e. the determination of the position of a point in a plane by its distance from two fixed lines. He did not himself discover all the power of this method, but he did enough to make further progress easy.

Philosophy is one of those subjects, like astrophysics and neurosurgery, that are not for the fainthearted. To delve into the absolutes of the human experience, to seek to advance the progress of enlightenment first expounded by the likes of the revered Aristotle and Plato, to search for the answers to the profound questions of the universe, often at the risk of deadly reprisal from entrenched powers, requires not only brilliance and tenacity but a deep sense of purpose. But even among this select fraternity, [René] Descartes stands out. From him did we get practical discoveries like coordinates in geometry and the law of refraction of light. But what he really did was to shake loose the human mind from the shackles of centuries of stultifying religious orthodoxy by creating an entirely original approach to reasoning: the Cartesian method.

Foundational Principles: Nature is lazy, it likes to copy. Everything is information. Information is not stuff, it is relationships. All behaviors are constrained by relationships. All behaviors are emergent. Every engine takes advantage of a difference. Everything is an approximation of something else. Ratio may be the only thing that is discrete. The bending of spacetime is a variation of scale. Behavior is built from a quantum of action in a field. If dimensions are virtual in the same way that the dimensions of consciousness are virtual then the density of information in a field will affect the scale or bending of spacetime. Gravity and scale are related. Gravity and information are related. Information and scale are related. If it's relational, there's a geometry involved. Truth as a scale coordinate; truth lives in the macro world, the micro world is uncertain. Truth as a time coordinate; truth lives in the past, the future is uncertain. Information from the micro future is formed into a macro past. The process of formation involves entanglement. Coffee and cream; 1) separate, 2) complex, 3) homogeneous. Information appears to increase and then decrease.

Noncommutative geometry has turned up in several approaches to quantum gravity, including string theory, DSR, and loop quantum gravity. But none of these capture the depth of Conne's original conception, which he and a few mathematicians, mostly in France, continue to develop. The various versions of it that appear in other programs are based on superficial ideas, such as making the coordinates of space and time into noncommuting quantities. Conne's idea is much deeper; it is a unification at the foundations of algebra and geometry. It could only be the invention of someone who does not just exploit mathematics but thinks strategically and creatively about the structure of mathematical knowledge and its future.

The process of objectifying the world through the primordial intuition of "repetition in time" and "following in time" gains in generality by the construction of mathematics from the same primordial intuition, without reference to direct applicability. In this way man has a ready-made supply of unreal causal sequences at his disposal, just waiting for an opportunity to be projected into reality. One should bear in mind that in mathematical systems with no time coordinate, all relations in practical applications clearly become causal relations in time; e.g. Euclidean geometry when applied to reality shows a causal connection between the results of different measurements made by means of the group of rigid bodies. Needless to say, in the application of a mathematical system, in general, only a fraction of the elements and substructures finds their correspondence in reality; the remainder plays the role of and unreal "physical hypothesis." Similarly, even with a limited development of method, the observed sequences no longer consist exclusively of phenomena evoked by man himself (acts without any direct instinctive aim, but carried out solely to complete the causal system into a more manageable one). The simplest example is the sound image (or written symbol) of number as a result of counting, or the sound image (or written symbol) of number as a result of measuring (this example shows how infinitely many causal sequences can be brought together under the viewpoint of one single law of causality on the basis of a mapping the numbers through mathematical induction.)

Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar co-ordinates false. One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient.

by the time of Emperor Ashoka, in the middle of the third century bce, India had adopted the twelve signs of the originally Babylonian Zodiac, as well as Hellenistic spherical geometry, celestial coordinate systems and the basic trigonometry of chords, all of which had their genesis in ancient Greece.

The prime advantage of defining the geometry of the pebble's surface in terms of general coordinates is that we can then proceed to define tensors that are true in any coordinate system, tensors that describe both the pebble's curvature (the Riemann curvature tensor) and physics (the field equations, for example).

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.

Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates. Rightly understood, there exists only a difference of gradation between these two kinds of geometry, according as one gives more prominence to the figures or to the formulas. Analytic geometry which dispenses entirely with geometric representation can hardly be called geometry; synthetic geometry does not get very far unless it makes use of a suitable language of formulas to give precise expression to its results.

Neointuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time, as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness. This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new twooneness, which process may be repeated indefinitely; this gives rise still further to the smallest infinite ordinal ω. Finally this basal intuition of mathematics, in which the connected and the separate, the continuous and the discrete are united, gives rise immediately to the intuition of the linear continuum, i.e., of the “between”, which is not exhaustible by the interposition of new units and which can therefore never be thought of as a mere collection of units. In this way the apriority of time does not only qualify the properties of arithmetic as synthetic a priori judgments, but it does the same for those of geometry, and not only for elementary two- and three-dimensional geometry, but for non-euclidean and n-dimensional geometries as well. For since Descartes we have learned to reduce all these geometries to arithmetic by means of coordinates.