We shall see later that the possibilities inherent in this process occasioned a chicken-and-egg debate between Bertrand Russell and Henri Poincaré at the end of the 19 th century. We therefore turn to 19 th century examinations of the intelligibility of geometry. Among the geometrical aspects of physical space that Newton established is the statement of his first law: Every body preserves in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. In the axiomatic geometries that Hilbert put forward, the fundamental objects points, lines, planes are not defined. The only conclusion one can draw, said Poincaré, is that either light rays travel along straight lines and space is non-Euclidean or that space is Euclidean and light rays travel along curves. To those who decided to read the Elements carefully and see how the crucial terms are used, it became apparent that the account is both remarkably scrupulous in some ways and flawed in others. Those who did accept it, and they were very few before the 1860s, nonetheless may well have welcomed a better account than the one Bolyai and Lobachevskii provided.
For Poincaré, talk about what we might call ordinary geometry, the sense of space that we have prior to advanced instruction, is really about the ability we have to measure things. In its synthetic form the successes of projective geometry were largely confined to the simplification it brought to the study of conics—all non-degenerate conics the circle, ellipse, parabola, and hyperbola are projectively equivalent. Truths established about the objects of geometry are purely abstract and hypothetical, because there is no such thing, for example, as a perfect circle. In his Exposition du système du monde of 1796 see Book V, Ch. Prior to axiomatisations of the theory at the end of the 19 th century, point, line, and plane were undefined concepts, with an intuitive interpretation that allowed for a ready passage between projective and Euclidean geometry. A line can be defined similarly.
With the intuition that distance is the primitive concept comes a greater appreciation of motion, or at least the results of being able to move objects around without altering them. In this way, it became possible to apply geometrical ideas in novel settings and in novel ways. The nature of the knowledge that geometry provided was also a matter of some discussion. All this convinced both Bolyai and Lobachevskii that the new geometry could be a description of physical space and it would henceforth be an empirical task to decide whether Euclidean geometry or non-Euclidean geometry was true. He could therefore define a sphere with its centre at a given point as the collection of all points equidistant from a given point.
N Words Connected with History 61. It is resistant to the idea of taking distance as a fundamental concept, or to the idea of replacing statements in geometry by statements about numbers say, as coordinates , although it is not hostile to coordinate geometry being erected upon it. That is, it takes the straightness of the straight line and the flatness of the plane as fundamental, and appeals to the incidence properties just described. The projective group preserves straight lines, and any ordered triple of collinear points can be mapped to any ordered triple of collinear points, and the map that sends a given ordered triple of distinct points to another ordered triple of distinct points is unique, but there is no transformation in the group that can map a quadruple of four collinear points onto an arbitrary such quadruple. Relative to this sense of distance, one can say if, as a region is moved around, points in it remain the same distance apart or not. This is true of regions of the sphere, but not of all of it at once.
As a result all definitions, theorems, and proofs in projective geometry have a dual character. Heat, the gravitational effect of massive objects, all these distorting influences are things that can be allowed for, because they can be changed. One might conduct the experiment further away from any massive objects, in emptier regions of space. Two cultures: Essays in honour of David Speiser, Birkhäuser, 133—138. Russell was trying to establish the Kantian a priori by arguing that there is one fundamental geometry, which is projective geometry, and we have synthetic a priori knowledge of it see Griffin 1991 on Russell and Nabonnand 2000 on the controversy.
But there is another weakness in the Elements that is also worth noting, although it drew less attention, and this is the nature of the plane. The other, non-Euclidean geometry, was a new and challenging metrical geometry. Elkana eds , Boston Studies in the Philosophy of Science, Boston: Reidel, volume 37, 39—57. Indeed, one might say that an epistemological gap remains to this day in philosophy in the form of a distinction between empirical and a priori knowledge that is still widely recognised. Mathematicians should no longer need to abstract some fundamental intuitions from what they believe about physical space, such as the nature and properties of straight lines or circles, and seek to build a true geometry on the basis of some axiomatic expression of those intuitions. Let us say that a purely synthetic geometry is one that deals with primitive concepts such as straight lines and planes in something like the above fashion. Chief among these was the force of gravity, which mathematicians in the Cartesian tradition regarded as a mysterious, even unacceptable, concept when it was introduced, but which by the start of the 19 th century had been shown by Laplace to be capable of dealing well with all the known motions of the solar system.
We have done this for our sense of distance on the surface of the Earth, and we can do this whether or not we also have some rigid body motions. Straight lines arise almost always as finite segments that can be indefinitely extended, but, as many commentators noted, although Euclid stated that there is a segment joining any two points he did not explicitly say that this segment is unique. To be demonstrated with complete rigour they must be considered as holding of bodies in a state of abstract perfection that they do not really have. On the other hand, the mere existence of formulae would not suffice to make them geometrical in character. Familiar congruence theorems follow in each edition until the parallel postulate could no longer be ignored.
This insight was made clear and explicit by Klein in a number of papers in the early 1870s. All this contributed to the central importance attributed to a non-metrical geometry based on little more than the concept of the straight line and on the incidence properties of lines and planes. Moreover, it was their origin in perception that gave these concepts their significance for science. It's working out very well. If there could be another way to define geometry, one that would lead to these formulae in various cases, the way would be open to rethink all of the questions about geometry that critical examination had opened.
To make matters worse, space is three-dimensional when regarded as made up of points, but four-dimensional when made up of lines. The sides of the triangle are, shall we say, made by rays of light. One line of enquiry led to geometries that emphasised straightness as the fundamental property typically, projective geometry and the other to geometries that emphasised the shortest aspect. Its proponents did not offer a sceptical conclusion. Analytic statements are a priori, the contentious class of a priori non-analytic statements contains those that could not be otherwise and so provide certain knowledge.
In his An Essay Concerning Human Understanding 1690 Locke asserted that When we possess ourselves with the utmost security of the demonstration, that the three angles of a triangle are equal to two right ones, what do we more but perceive, that equality to two right ones does necessarily agree to, and is inseparable from, the three angles of a triangle? He then showed how to define a plane by capturing the intuition that given two distinct points a plane is the collection of points in space that are the same distance from each of the two given points. Next, he checked that this intrinsic property of curvature persisted in higher dimensions, which it does. Lobachevskii argued firstly that geometry was a science of bodies in space, and that space is three-dimensional. The hundred greatest mathematicians of the past this is the long page with list and biographies click here for just the list with links to the biographiesor click here for a list of the 200 greatest of all time. Two bodies not in contact are separated and a suitable third body in contact with both of them measures the distance between them, a concept that was otherwise undefined.