ONT Re: Manifolds of Sensuous Impressions
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
MSI. Note 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| 11. On the Hypotheses Which Lie at the Basis of Geometry (cont.)
|
| Riemann opened his lecture with the following much-quoted passage.
|
| | It is known that geometry assumes, as things given, both the notion of space and
| | the first principles of constructions in space. She gives definitions of them
| | which are merely nominal, while the true determinations appear in the form of
| | axioms. The relation of these assumptions remains consequently in darkness;
| | we neither perceive whether and how far their connection is necessary, nor,
| | 'a priori', whether it is possible.
|
| "From Euclid to Legendre", Riemann declares, nothing
| has been done to remove this obscurity. He continues:
|
| | The reason of this is doubtless that the general notion of multiply extended
| | magnitudes (in which space-magnitudes are included) remained entirely unworked.
| | I have in the first place, therefore, set myself the task of constructing the
| | notion of a multiply extended magnitude out of general notions of magnitude.
| | It will follow from this that a multiply extended magnitude is capable
| | of different measure-relations, and consequently that space is only a
| | particular case of a triply extended magnitude. But hence flows as a
| | necessary consequence that the propositions of geometry cannot be derived
| | from general notions of magnitude, but that the properties which distinguish
| | space from other conceivable triply extended magnitudes are only to be deduced
| | from experience. Thus arises the problem, to discover the simplest matters of
| | fact from which the measure-relations of space may be determined; a problem
| | which from the nature of the case is not completely determinate, since there
| | may be several systems of matters of fact which suffice to determine the
| | measure-relations of space -- the most important system for our present
| | purpose being that which Euclid has laid down as a foundation. These
| | matters of fact are -- like all matters of fact -- not necessary, but
| | only of empirical certainty; they are hypotheses. We may therefore
| | investigate their probability, which within the limits of observation
| | is of course very great ...
|
| He then proceeded to consider separately the notion of n-ply extended magnitude,
| and of the measure relations possible in such a manifold. In explicating the
| former Riemann states:
|
| | Magnitude-notions are only possible where there is an antecedent
| | general notion which admits of different specialisations. According
| | as there exists among these specialisations a continuous path from one
| | to another or not, they form a 'continuous' or 'discrete' manifoldness:
| | the individual specialisations are called in the first case points,
| | in the second case elements, of the manifoldness.
|
| As examples of notions whose specializations
| form a continuous manifoldness Riemann offers
| positions abd colors. He then continues:
|
| | Definite portions of a manifoldness, distinguished by a mark
| | or by a boundary, are called Quanta. Their comparison with
| | regard to quantity is accomplished in the case of discrete
| | magnitudes by counting, in the case of continuous magnitudes
| | by measuring. Measure consists in the superposition of the
| | magnitudes to be compared; it therefore requires a means
| | of using one magnitude as the standard for another.
|
| (MGM, pp. 219-220)
|
| Murray G. Murphey,
|'The Development of Peirce's Philosophy',
| first published, Harvard University Press, Cambridge, MA, 1961.
| reprinted, Hackett Publishing Company, Indianapolis, IN, 1993.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o