Dasein - Monoskop

128 PART III
manifolds are constructs, built up by mathematical operations from
somethings-in-general: the number two, for instance, is 'something
and something'; a set consists of somethings; and so on. Now
Husserl's point is that just as a theory form correlates with a manifold,
so also formal apophansis correlates with formal ontology. Judgments
are always judgments about something, some object or objectivity,
and all objects "have being for us—as truly existent or possibly
existent modes—only as making their appearance in judgments.
Accordingly, in all formal distinctions pertaining to judgments, differences
among object-forms are included . .." 507 For example, in
formal apophansis we have the distinction between singular and plural,
and in formal ontology the distinction between 'something' and
'somethings'.
What this distinction allows Husserl to do is to follow Hilbert's
formalistic program without falling an easy prey to the charge of formalism:
even though formal mathematics, that is, formal apophansis,
operates 'merely' syntactically, its signs are not void of meaning:
their meanings are given by the categories of formal ontology that
are correlated with them. Husserl makes this point especially forcefully
when discussing theory form and manifold, that respectively
constitute the highest categories of formal apophansis and formal
logic. He opposes tendencies, "to put in the place of the real theory
of manifolds its symbolic analogue—that is, to define manifolds in
terms of mere game rules". 508 In the definition of a manifold, Husserl
claims, we are not to speak of how we are allowed or supposed to
manipulate signs, rather we are to speak of connecting forms holding
among "the objects belonging to the manifold (conceived at first as
only empty somethings, 'objects of thinking')". 509 However, once we
accept this façon de parler, we are free to ignore Frege's warning
against formalism, and free to follow Hilbert:
... it is understandable that, for a consciously or unconsciously
*pure 9 formal mathematicsf there can be no cognitional consideration
other than those of non-contradiction"; ...
It is otherwise, to be sure, for the logician: ... he will not easily
come upon the thought of making this reduction to an analytics
of pure senses; and therefore he will acquire mathematics as
only an amplified logic, which, as a logic, relates essentially to