Dasein - Monoskop

44 PART II
the debate between Frege and Hilbert had a fascinating "deeper dimension".
Their exchange over the question of implicit definitions
was very much a debate over which view of logic we have to adopt:
logic as language or logic as calculus. For example, Hilbert allowed
for the free interpretability of his axioms. Furthermore, metalogical
proofs—not allowed according to FYege's view—form the heart
of Hilbert's mathematical thinking. That Husserl was siding with
Hilbert can both be seen from his adoption of Hilbert's ideas and
also be read more directly from Husserl's notes on the correspondence
between Hilbert and Frege. Having extracted Frege's main
arguments, Husserl writes:
Frege does not understand the meaning of the Hilbertian "axiomatic"
founding of geometry, namely that it is a purely formal
system of conventions, whose theory form is equal to the Euclidian.
When can we be sure that we have not drawn the matter of
some region of knowledge into the deductions and that we have
inferred purely logically? ... only when we grasp the matter
symbolically, when we rise to a formal system, to a theory form,
which is defined by the sentence forms of the material principles
of the region. ... The axiomatic foundations define the formal
region.
no
It was also under the influence of Hilbert that from 1901 onwards
Husserl reformulated his earlier program for handling the extension
problem. 133 Now Husserl no longer suggested tackling this problem
by dropping restrictions within a calculus (for instance, the restriction
a > b for the operation a - 6), but rather by expanding an
axiomatic system where one axiomatic system A is an expansion of
another axiomatic system B if and only if all theorems of B are also
theorems of A but not vice versa. Husserl did not work out this program
in detail, but he did work intensively on the problem of defining
the conditions under which a theorem of B containing only expressions
of A is valid. The attention given by Husserl to this problem is
noteworthy because in working out a solution to the problem Husserl
shows an interest in metalogical questions and proofs that Frege—at
least initially—was unable to consider in a systematic way because