ONT Re: Manifolds of Sensuous Impressions
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MSI. Note 3
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| 11. On the Hypotheses Which Lie at the Basis of Geometry (cont.)
|
| Riemann then shows how an n-ply extended manifold
| may be constructed and determination of place in it
| reduced to determinations of quantity. He then faces
| the question of the measure relations possible in such
| a manifold. Mathematically, this portion of the essay is
| of great significance, but the technical development need
| not concern us here. The basic idea may be summarized as
| follows. Riemann notes that measurement requires quantity
| to be independent of place and he accordingly adopts the
| hypothesis that the length of lines is independent of their
| position so that every line is measurable by every other.
| If we define distance as the square root of a quadratic
| function of the coordinates then Riemann shows that for
| the length of a line to be independent of its position,
| the space in which the line lies must have constant
| curvature. "The common character of these continua
| whose curvature is constant may also be expressed
| thus, that figures may be moved in them without
| stretching ... whence it follows that in
| aggregates with constant curvature
| figures may have any arbitrary
| position given them."
|
| In the final section of the essay Riemann turns to the
| question of the application of his technical apparatus
| to empirical space for the determination of its metric
| properties. In a space of constant curvature in which
| line length is independent of position, the empirical
| truth of the Euclidean axiom that the sum of the angles
| of a triangle is equal to two right angles is sufficient
| to determine the metric properties of that space. But such
| empirical determinations run into difficulty in the cases of
| the infinitely great and the infinitely small. "The questions
| about the infinitely gerat are for the interpretation of nature
| useless questions", according to Riemann, but the same is not true
| on the side of the infinitely small. He continues:
|
| | If we suppose that bodies exist independently of position,
| | the curvature is everywhere constant, and it then results
| | from astronomical measurements that it cannot be different
| | from zero; or at any rate its reciprocal must be an area
| | in comparison with which the range of our telescopes may
| | be neglected. But if this independence of bodies from
| | position does not exist, we cannot draw conclusions
| | from metric relations of the great, to those of the
| | infinitely small; in that case the curvature at
| | each point may have an arbitrary value in three
| | directions, provided that the total curvature
| | of every measurable portion of space does not
| | differ sensibly from zero. ... Now it seems
| | that the empirical notions on which the metrical
| | determinations of space are founded, the notion of
| | a solid body and a ray of light, cease to be valid
| | for the infinitely small. We are therefore quite
| | at liberty to suppose that the metric relations
| | of space in the infinitely small do not conform
| | to the hypotheses of geometry; and we ought in
| | fact to suppose it, if we can thereby obtain a
| | simpler explanation of phenomena.
| |
| | The question of the validity of the hypotheses of
| | geometry in the infinitely small is bound up with
| | the question of the ground of the metric relations
| | of space. In this last question ... is found the
| | application of the remark made above; that in a
| | discrete manifoldness, the ground of its metric
| | relations is given in the notion of it, while
| | in a continuous manifoldness, this ground
| | must come from outside. Either therefore
| | the reality which underlies space must
| | form a discrete manifoldness, or we
| | must seek the ground of its metric
| | relations outside it, in binding
| | forces which act upon it.
|
| But the final answer to this question, Riemann asserts,
| must come from physics rather than from pure mathematics.
|
| MGM, pp. 221-222.
|
| Murray G. Murphey,
|'The Development of Peirce's Philosophy',
| first published, Harvard University Press, Cambridge, MA, 1961.
| reprinted, Hackett Publishing Company, Indianapolis, IN, 1993.
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