FOM: Carnap on geometry

[Neil Tennant]

My machine went down last night, and I think I might have missed out
on some of the replies to Mic Detlefsen's posting on the way the three
'isms' (logicism, formalism, intuitionism) were responses to the
discovery of non-Euclidean geometries. I found his insights both
fascinating and illuminating.

It might be worth adding that Carnap---who was once Frege's student in
Jena---wrote a doctoral dissertation titled "Der Raum". In it Carnap
distinguishes three kinds of geometry: formal, intuitive (or
phenomenal), and physical geometry. [German: 'Wir unterscheiden den
formalen, den Anschauungsraum, und den physischen Raum.'] There is
nothing usefully synthetic a priori about formal geometry, since it is
essentially uninterpreted ['Bei ihm handelt es sich also ... um
bedeutungslose Beziehungsst"ucke.']. So one has to look to phenomenal or
to physical geometry if one seeks to locate might what survive of the
geometric synthetic a priori in the wake of the discovery of
non-Euclidean geometries.

Phenomenal geometry, according to Carnap, has to do with the
properties of spatial figures whose particular peculiarities we grasp
from the opportunity of sensory experience or even [the] mere
[exercise of our] imagination ['deren bestimmte Eigenheit wir bei
Gelegenheit sinnlicher Wahrnehmung oder auch blosser Vorstellung
erfassen']. So it deals not with the spatial facts of experiential
reality ['die in der Erfahrungswirklichkeit ... r"aumlichen
Tatsachen'], but only with the essential nature or being of the
structures themselves, which can be recognized in any [number of]
representatives of their kind ['das >>Wesen<< jener Gebilde selbst,
das an irgendwelchen Artvertretern erkannt werden kann.']

Carnap thinks that the synthetic a priori survives in the various
phenomenal or physical geometries (projective, affine, Euclidean,
Riemannian). It is to be found precisely in the *topological
statements* about inclusion and contact among spatial structures
['f"ur alle topologischen Aussagen und nur f"ur diese, d.h. f"ur die
Aussagen "uber Ineinanderliegen und Zusammenhang der Raumgebilde'].

One's knowledge of physical geometry presupposes that of phenomenal
geometry. The latter in turn finds the pure form of its structure
['die reine Form seines Gef"uges'] foreshadowed or prefigured in
formal geometry, and therefore involves the latter as a 'thought-
constraining presupposition' ['und hat ihn daher zur denkm"assigen
Voraussetzung'].

What I would like to know from any fom-ers with an interest in such
matters is whether it is the case that all geometries share some
common axiomatized core for the topologies of their spaces. It is a
seductive thought for a neo-Kantian that there might be at least
*some* constraints on the spatial possibilities for experience. The
metric might well be conventional or a posteriori: but perhaps one's
sense of in-and-out, of touching, of overlap etc. is common to all
geometries, and neither mutilated nor sacrificed in any in the
transition from Euclidean to any of the great many non-Euclidean
geometries?

If so, perhaps Carnap was onto something here.

Neil Tennant