Ramble, Part 45: Making Models
Feb. 2nd, 2008 05:19 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThough Gauss, Bolyai, and Lobachevsky had been unable to find contradictions in hyperbolic geometry, the question still remained whether there actually were any such contradictions. The same was true of the elliptic geometry studied by Gauss's student Bernhard Riemann. (In that geometry, Saccheri's "hypothesis of the obtuse angle", there are no parallels, and lines have finite length.) The question of how to show that the new geometries were not self-contradictory was dealt with in the next generation or so, most notably by Henri Poincaré.
Clearly the assumptions of these new geometries were not true of the world in which mathematicians had been working. Was there, though, something else of which they were true? If that could be shown to be the case, that would suffice to show that they were not self-contradictory.
One avenue might be to look at surfaces other than the plane. These had undergone intensive scrutiny by, among others, Gauss; geodesics, shortest paths on a surface, could be recruited to take the place of lines. This looked promising. On one of the best-known surfaces, the sphere itself, geometry was well-understood, and as far back as Lambert the connection between the hypothesis of the obtuse angle and spherical geometry had been noticed. There was only one disruptive problem. Geodesics on a sphere are great circles - circles whose center is the center of the sphere, such as meridians of longitude (but not, except for the equator, parallels of latitude) - and any two great circles intersect, not in one, but in two points.
Poincaré took a different tack. In hyperbolic geometry, we are discussing "points" and "lines", and the latter may be defined to be shortest paths if we only have a notion of distance. But what are these points and lines, and this notion of distance? Clearly not the usual notions. Perhaps, if we choose different notions of "point" and "distance", these axioms may be made to fit. Poincaré, indeed, found two different (but, he showed, essentially equivalent) ways of doing this; I'll describe one of them, the "Poincaré half-plane".
In this construction, "points" are points on the usual Cartesian plane, with the restriction that the y-coordinate must be positive. The notion of distance is a bit abstruse, so I'll relegate it to a blockquote, but the notion of "line" it gives rise to is this. A line, in the Poincaré half-plane, is either a vertical ray - a set of the form {(a,y): y>0}, where a is some constant - or else a (Euclidean) semicircle whose center lies on the x-axis. One can then think of a hyperbolic line as having two "endpoints" (which are not, I emphasize, part of the half-plane) - the points on the x-axis in the semicircular case, or the point (a,0) and a "point at infinity", infinitely far away in the vertical direction, in the case of a vertical ray. Two lines are parallel, in the sense I gave in the last ramble, if they share an endpoint, and ultraparallel if they share neither points in the half-plane nor endpoints.
Elliptic geometry may be handled in a similar fashion. If spherical geometry fails as a model because its lines intersect in a pair of diametrically opposite points, why, assign to "point" the meaning "pair of diametrically opposite points on a sphere", and adjust the notion of distance as needed. The axioms of the hypothesis of the obtuse angle are satisfied, and hence they are consistent.
Almost. The models I've been describing sit inside Euclidean geometry; that is, the meanings assigned to "point" and "distance" are Euclidean concepts (if not the Euclidean concepts that usually go by those names), and hence they are consistent if Euclidean geometry is. An obstreporous student might ask: how do we know that Euclidean geometry is consistent? This would be a very good question, and we'll talk about it at some point in the future.
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Clearly the assumptions of these new geometries were not true of the world in which mathematicians had been working. Was there, though, something else of which they were true? If that could be shown to be the case, that would suffice to show that they were not self-contradictory.
One avenue might be to look at surfaces other than the plane. These had undergone intensive scrutiny by, among others, Gauss; geodesics, shortest paths on a surface, could be recruited to take the place of lines. This looked promising. On one of the best-known surfaces, the sphere itself, geometry was well-understood, and as far back as Lambert the connection between the hypothesis of the obtuse angle and spherical geometry had been noticed. There was only one disruptive problem. Geodesics on a sphere are great circles - circles whose center is the center of the sphere, such as meridians of longitude (but not, except for the equator, parallels of latitude) - and any two great circles intersect, not in one, but in two points.
Poincaré took a different tack. In hyperbolic geometry, we are discussing "points" and "lines", and the latter may be defined to be shortest paths if we only have a notion of distance. But what are these points and lines, and this notion of distance? Clearly not the usual notions. Perhaps, if we choose different notions of "point" and "distance", these axioms may be made to fit. Poincaré, indeed, found two different (but, he showed, essentially equivalent) ways of doing this; I'll describe one of them, the "Poincaré half-plane".
In this construction, "points" are points on the usual Cartesian plane, with the restriction that the y-coordinate must be positive. The notion of distance is a bit abstruse, so I'll relegate it to a blockquote, but the notion of "line" it gives rise to is this. A line, in the Poincaré half-plane, is either a vertical ray - a set of the form {(a,y): y>0}, where a is some constant - or else a (Euclidean) semicircle whose center lies on the x-axis. One can then think of a hyperbolic line as having two "endpoints" (which are not, I emphasize, part of the half-plane) - the points on the x-axis in the semicircular case, or the point (a,0) and a "point at infinity", infinitely far away in the vertical direction, in the case of a vertical ray. Two lines are parallel, in the sense I gave in the last ramble, if they share an endpoint, and ultraparallel if they share neither points in the half-plane nor endpoints.
Recall the usual definition of arc-length in the plane. A curve is given by a pair of functions x=f(t), y=g(t), where t, which may be thought of as time, ranges between two values a and b. The usual "distance element" ds is dt times the square root of f'(t)2+g'(t)2, and the length of the curve is obtained by integrating the distance element between a and b. In the Poincaré half-plane, the distance element is the usual one divided by y. That is, vertical motion grows harder, as it were, as one approaches the x-axis, and easier as one moves away. The shortest paths, then, are as described above. (As an example: given two points a+bi and a+ci on the same vertical ray, use the path given by f(t)=a, g(t)=b+t(c-b), where t ranges between 0 and 1. Integration yields the distance between these points, |ln(b/c)|.)This is an example of a "model" for the axioms of hyperbolic geometry. The fundamental concepts are "point" and "distance", which (in the axioms) are undefined; a model assigns specific meanings to these concepts, in such a way as to make the axioms true. This is enough to show that the axioms of hyperbolic geometry are consistent.
Elliptic geometry may be handled in a similar fashion. If spherical geometry fails as a model because its lines intersect in a pair of diametrically opposite points, why, assign to "point" the meaning "pair of diametrically opposite points on a sphere", and adjust the notion of distance as needed. The axioms of the hypothesis of the obtuse angle are satisfied, and hence they are consistent.
Almost. The models I've been describing sit inside Euclidean geometry; that is, the meanings assigned to "point" and "distance" are Euclidean concepts (if not the Euclidean concepts that usually go by those names), and hence they are consistent if Euclidean geometry is. An obstreporous student might ask: how do we know that Euclidean geometry is consistent? This would be a very good question, and we'll talk about it at some point in the future.
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