Manifold (Mannigfaltigkeit)
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Edmund Husserl, Formal and Transcendental Logic: 28-32
Edmund Husserl, Logical Investigations (Prolegomena): 70
 
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The objective correlate of the concept of a possible theory, determined only in its form, is the concept of any possible province of cognition that would be governed by a theory having such a form. (Edmund Husserl, Logical Investigations (Prolegomena): s. 19)

Manifolds are thus in themselves compossible totalities of objects in general, which are thought of as distinct only in empty, formal generality and are conceived of as defined by determinate modalities of the something-in-general. Among these totalities the so-called “definite” manifolds are distinctive. Their definition through a “complete axiomatic system” gives a special sort of totality in all deductive determinations to the formal substrate-objects contained in them. With this sort of totality, one can say, the formal-logical idea of a “world-in-general” is constructed. The “theory of manifolds” in the special sense is universal science of the definite manifolds”. (Edmund Husserl, The Crisis of European Philosophy: 45-46)

...once we have worked out the laws governing mathematical manifolds of a certain sort, our results can be applied - by a process of 'specialisation' - to every individual manifold sharing this same form. Husserl's discovery of this essential community of logic and ontology is of the utmost importance for his philosophy of mathematics. It can be shown to imply a non- trivial account of the applicability of mathematical theories - of a sort that is missing, for example, from a philosophy of mathematics of the kind defended by Frege - as a matter of the direct specialisation of the relevant formal object-structures to particular material realisations in given spheres. (Smith, Barry, Logic and Formal Ontology: Section 5)

As Husserl himself points out, certain branches of mathematics are partial realisations of the idea of a formal ontology. The mathematical theory of manifolds as this was set forth by Riemann and developed by Grassmann, Hamilton, Lie and Cantor, was to be a science of the essential types of possible object-domains of scientific theories, so that all actual object- domains would be specialisations or singularisations of certain manifold-forms. And then: 'If the relevant formal theory has actually been worked out in the theory of manifolds, then all deductive theoretical work in the building up of all actual theories of the same form has been done. (Smith, Barry, Logic and Formal Ontology: Section 5)

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