Logic is not, however, concerned only with meanings and with associated instantiating acts. For even a deductively closed collection of meanings will constitute a science only where we have an appropriate unity and organisation also on the side of the objects to which the relevant acts refer. The unity of scientific theory can in fact be understood to mean either (1) an interconnection of truths (or of propositional meanings in general), or (2) an interconnection of the things to which our cognitive acts are directed.
Since meanings are just ways of being directed towards objects, it follows that (1) and (2) ‘are given together a priori and are mutually inseparable’ (I A228f./225). And logic, accordingly, relates not only to meaning categories such as truth and proposition, subject and predicate, but also to object categories such as object and property, relation and relatum, manifold, part, whole, state of affairs, and so on.(18) Logic seeks therefore to delimit the concepts which belong to the idea of a unity of theory in relation to both meanings and objects, and the truths of logic are all the necessary truths relating to those categories of constituents, on the side of both meanings and objects, from out of which science as such is necessarily constituted.
Husserl’s conception of the science of logic as relating also to formal-ontological categories such as object, state of affairs, unity, plurality, and so on, is not an arbitrary one. These concepts are, like the concepts of formal logic, able to form complex structures in non-arbitrary, law-governed ways, and they, too, are independent of the peculiarity of any material of knowledge. This means that in formal ontology, as in formal logic, we are able to grasp the properties of given structures in such a way as to establish in one go the properties of all formally similar structures.
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.’ (I A249f./242) That is to say, 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.
How, then, are we more precisely to understand Husserl’s account of the relation between theory as structure of meanings and theory as structure of objects and objectual relations? A theory as a structure of meanings is a certain deductively closed combination of propositions (and higher order meaning structures) which are themselves determinate sorts of combinations of concepts and combination-forms. Just as the propositions are species of judgments, so the concepts which are their parts are species of linguistically expressible presentations. The concepts in question are in each case of determinate material: they are concepts of a dog, of an electron, of a colour (or of this dog, of dogs in general, of electrons in general) and so on. But we can move from this level of material concepts to the purely formal level of: a something, this something, something in general and so on, by allowing materially determinate concepts to become mere place-holders for any concepts whatsoever – a process of ‘formalisation’. The idea of a theory-form now arises when we regard all materially determinate concepts in a given body of theory as having been replaced in this fashion by mere variables, by materially empty concepts, so that only the formal structure of the theory is retained.(19)
What, now, is the objectual correlate of such a theory-form? It is the structure shared in common by all possible regions of knowledge to which a theory of this form can relate, a structure determined solely as one ‘whose objects are such as to permit of these and these connections which fall under these and these basic laws of this or that determinate form.’ (I A248/241) Here again, therefore, it is form alone that serves as determining feature. The objects in the given structure are quite indeterminate as regards their matter: they constitute, as it were, mere shells or frames into which various matters can, in principle, be fitted. And the structure as a whole is determined merely by the fact that its objects (nodes) stand in certain formally determined relations and permit of certain formal operations, for example the operation that is represented by ‘+’, defined as commutative, associative, etc.
For a collection of scientific statements to constitute a theory, then, there must be on this purely formal level an ‘ideal-lawful adequacy of its unity as unity of meaning to the objective correlate meant by it’ (II A92/323). The objects meant by the constituent propositions of the theory (and therefore also by corresponding judging acts) must hang together in a precisely appropriate way, must constitute the formal unity of a certain determinate formal manifold.
Barry Smith, Logic and Formal Ontology