The world has a certain sensible, material stuff. Within this stuff we can, if we strain our mental eyes, pick out certain categorial objects. By means of suitable acts of relating or of setting into relief we can make out certain higher order formally determined structures and we can carve out for ourselves new objects by cleaving the relevant matters along formally determined contour lines.39 The material stuff of the world thereby serves as immediate foundation for the categorially shaped objects which result.
This process can however be carried forward. The operations involved in categorial shaping can be iterated, so that the objects of categorial acts are themselves subjected to further categorial shaping of higher order:
categorial unities may again and again become the objects of new connecting, relating or ideating acts. Thus for example universal objects can be collectively connected, the collections thereby formed then again connected with other collections of similar or different type, and so on in infinitum. (II A653/816)
The resulting higher order categorial acts can indeed be such that the sensory material with which we started is no longer present even in a subsidiary way in the contents of the acts in question. This is the sense of Husserl’s designation of the categorial disciplines as ‘pure’:
Like the whole of pure logic, so all pure arithmetic, the pure theory of manifolds, in short the pure mathesis in the all- embracing sense, are pure in the sense that they contain no sensuous concept in their entire theoretical fabric. (II A656/819)
Categorial shaping is a purely intellectual matter. But the objects it picks out are not denizens of any separate, purely intellectual realm. It is, rather, as if these objects sit on top of the perceptual world in such a way as to leave all the real, sensory structures and all the real unities which lie beneath them unaffected.40 Thus categorial forms do not glue, tie or put real parts together so that new sensuously perceivable wholes would emerge. The relating and connecting, the setting into focus and the drawing of boundaries that is involved in categorial processing merely sets up a new view [Fassung] of what is intuited on the primary level of sensory acts, a view which ‘can be given only in such a founded act, so that the [Platonistic] idea of a straightforward perception of that which has been formed or of a givenness through some other straightforward intuition, is absurd.’ (II A658/820) From this it follows however that – as far as concerns the world of what happens and is the case – categorial shaping leaves everything as it is.
It is not, however, as if the categorially perceiving intellect enjoys complete freedom in his forming and shaping:
The very fact that the categorial forms constitute themselves in founded characters of acts, and in these alone, involves a certain necessity of connection. How, otherwise, could we speak of categorial perception and intuition, if any conceivable matter could be put into any conceivable form, i.e. if the founding straightforward intuitions permitted themselves to be arbitrarily connected together with categorial characters? (II A660/821)
The insistence on the possibility of fulfilment – in fact of a complex chain of fulfilments leading back, ultimately, to sensory intuitions – imposes quite determinate constraints on the shaping that is possible on the higher categorial levels. Indeed there are laws governing the possibility and impossibility of combination and iteration of categorial operations that are analogous to the laws governing the combination of meanings on the level of pure grammar, laws having their origins in what is possible and impossible in virtue of the compatibilities among acts of identification, collection, setting into relief, and so on, in relation to given foundations. We cannot convert a part- whole relationship into a relationship of discrete items and preserve the possibility of adequate fulfilment. Such examples point to a family of laws governing the transformation of meanings, for example from ‘w is a part of W‘ to ‘W is a whole relative to w’, from ‘a certain A is B‘ to ‘not all A’s are not B’s’, and so on, in such a way that the possibility of fulfilment is preserved.41 Because the species of material foundations hereby involved ‘are quite freely variable and are only subject to the obvious ideal condition of capacity to function as carriers of the relevant forms, the laws in question are of an entirely pure and analytic character’ (II A661/822). They hold in abstraction from all sensuous stuff, ‘and are accordingly not capable of being affected at all by limitless variation of such stuff’ (II A672/831). Hence they do not need grounding in experience, and it is senseless to suppose that the world might somehow fail to satisfy them: ‘Laws which refer to no fact cannot be confirmed or refuted by fact.’
There is need of no metaphysical or other sorts of theories to explain the harmony of the course of nature and the ‘inborn’ laws of ‘understanding’: instead of an explanation one needs merely a phenomenological clarification of meaning, thinking and knowing and of the ideas and laws which spring from these. (II A671f./830)
In the theoretical domain of logic proper we are concerned exclusively with authentic thinking, with cognitive acts and processes capable of corresponding in the full sense to objects, i.e. of being bound up ‘with an intuition which fulfils them totally and singly’ (II A666/826). But there are of course free and easy categorial acts which are a matter of mere signitive directedness to categorial objects already constituted. Indeed we might entirely abandon the insistence on fulfilment, and rest content with an empty categorial forming and shaping, a forming and shaping which does not understand itself and which can take place fully ad libitum. We could then talk purely signitively about (even build entire axiom systems relating to) the square root of Napoleon’s hat, or the part-whole relations between Wellington’s boots and the mother of my umbrella. For in the sphere of inauthentic thinking, of mere signification, ‘we are free of all constraint by categorial laws. Here anything and everything can be brought together in unity.’ (II A666/826)
Barry Smith, Logic and Formal Ontology