Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
William Kingdon CliffordRelated topics
curvature different experience experiment explanation find geometry help live number paper particular phenomenon physical plane reason sheet shew small solid space surface three total yet zero riemann linesRelated quotes
Like all great churches, that are not mere store-houses of theology, Chartres expressed, besides whatever else it meant, an emotion, the deepest man ever felt,- the struggle of his own littleness to grasp the infinite. You may, if you like, figure in it a mathematic formula of infinity,- the broken arch, our finite idea of space; the spire, pointing, with its converging lines, to Unity beyond space; the sleepless, restless thrust of the vaults, telling the unsatisfied, incomplete, overstrained effort of man to rival the energy, intelligence and purpose of God. Thomas Aquinas and the schoolmen tried to put it in words, but their church is another chapter. In act, all man's work ends there;- mathematics, physics, chemistry, dynamics, optics, every sort of machinery science may invent,- to this favor come at last, as religion and philosophy did before science was born.
Henry Adams
Many is the mirage I chased. Always I was overreaching myself. The oftener I touched reality, the harder I bounced back to the world of illusion, which is the name for everyday life. 'Experience! More experience!' I clamored. In a frantic effort to arrive at some kind of order, some tentative working program, I would sit down quietly now and then and spend long, long hours mapping out a plan of procedure. Plans, such as architects and engineers sweat over, were never my forte. But I could always visualize my dreams in a cosmogonic pattern. Though I could never formulate a plot I could balance and weigh opposing forces, characters, situations, events, distribute them in a sort of heavenly lay-out, always with plenty of space between, always with the certitude that there is no end, only worlds within worlds ad infinitum, and that wherever one left off one had created a world, a world finite, total, complete.
Henry Miller
It is quite easy to include a weight for empty space in the equations of gravity. Einstein did so in 1917, introducing what came to be known as the cosmological constant into his equations. His motivation was to construct a static model of the universe. To achieve this, he had to introduce a negative mass density for empty space, which just canceled the average positive density due to matter. With zero total density, gravitational forces can be in static equilibrium. Hubble's subsequent discovery of the expansion of the universe, of course, made Einstein's static model universe obsolete. ...The fact is that to this day we do not understand in a deep way why the vacuum doesn't weigh, or (to say the same thing in another way) why the cosmological constant vanishes, or (to say it in yet another way) why Einstein's greatest blunder was a mistake.
Frank Wilczek
We had a very respectable force, as to numbers; between eight and nine thousand, rank and file, upon the ground; but of these we attempted to select a particular corps to possess ourselves of the enemy's lines, partly by force, and partly by stratagem; but w r e could not make up the necessary number that was thought sufficient to warrant the attempt, which was five thousand, including the Continental and State troops. This body was to consist of men, who had been in actual service before, not less than nine months. However, the men were not to be had, and, if they could have been found, there was more against it than for it. Colonel Laurens was to have opened the passage by landing within the enemy's lines, and getting possession of a redoubt at the head of Easton's beach. If we had failed in the attempt, the whole party must have fallen a sacrifice, for their situation would have been such that there was no possibility of getting off.
Nathanael Greene
...the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous, although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i.e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. ...such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.
Hans Reichenbach
