Mic Detlefsen wrote:
[An unresolved question for our understanding of H's views on geometry is
exactly how he conceived of the real/ideal distinction in geometry. He
sometimes used terms like 'Schulgeometrie' and 'fundamental geometry' that
suggest he saw a certain part of geometry as corresponding to 'real'
geometry. However, to my knowledge, he never provided (or attempted to
provide) a demarcation of real and ideal methods in geometry that was
comparable in precision and motivation to the real/ideal distinction he
drew for the foundations of arithmetic.]
How can we resolve this question of the real/ideal distinction in
Hilbert?
Two proposals.
[1] Specifically w.r.t. GG.
Schulgeometrie, fundamental geometrie, real geometrie: this is
covered by Axiom Groups I(axs.1-3, planar, the rest non-planar))-IV and
Axiom V.1 in G.G. (FoG).
Familiarly (but see para. 9 of GG; this from FoG): "real geometry"
(as just specified) has as a model the field Omega that results from
closure of {the number 1} under the arithmetic operation of plus, times,
subtraction, multiplication, and |sqrrt of (1 + xexp2)|. That is, full
Dedekind continuity is not needed in "real geometry" (see Dedekind quote
below).
Think of Axiom V.2, the Axiom of LINE Completeness as analogous to
Lavosier's utilization of the Principle of Sufficient Reason, that is of
introducing ideal elements into "real geometry", viz., all the "real
numbers" not included above model.
We are so accepting of the correlation between the geometric linear
continuum and the field of real numbers that we forget that historically
this correlation was thought of as going beyond or idealizing, geometric
continuity. This attitude is sharply expressed in Dedekind in his debate
with Lipschitz (the following is from Chapter 7 of Bottazinni) [recall that
for Dedekind, the real numbers were an intellactual/mental construction,
and so ideal rather than real]. Dedekind (p.271 Botta):
"I can represent to myself all of space and every line in it as
discontinuous throughout and I hold that every man can do the same. [The
fact remains that the continuity of space is not] a postulate that is
inseprably tied to Euclid's geometry. Euclid's entire system remains even
without continuity -- a result that is surprising for many and to me
therefore well worth mention." (The issue is -- or this is revealed by the
fact that --that "real geometry" has the model in the algebraic numbers
indicated above).
Remark 1. this is interesting in the light of Weyl's witholding
from the complete Dedekind continuum.
Remark 2. The axiom of line completeness can be given an exact
Kantian treatment in the spirit of, e.g., Kant's treatment of the Principle
of Sufficient Reason, that is, they are both "optimizing" principles of
reasoning but do not (necessarily) reflect any features of nature.
[2] Clearly, reflections on the ideal/real distinction in geometry must
work through (ik) the paper in the Appendix to Hilbert's FoG itself titled
"Foundations of Geometry", and much more importantly,
Hilbert and Cohn-Vossen's Anschauliche Geomtrie.
Robert Tragesser