We now see which role quantitative synthesis plays within the complex act of figurative synthesis. We have seen that, combined with qualitative syn- thesis, quantitative synthesis allows us to represent manifolds of empirical representations. This is necessary for us to be able to represent the various aspects of empirical objects.
What has not become clear yet is the relationship between this act of quantitative synthesis and theschema, the categories and thelogical forms
of Quantity. In order to understand this relationship, we need to note that sensations are not the only kinds of representations the activity of quan- titative synthesis can be applied to. In 2.3.3, we saw that quantitative synthesis generates a form that enables us to represent a manifold of sen- sations. Longuenesse thinks this form can be ‘filled’ with sensations, but that it can be filled with other sensible representations too. Quantitative synthesis thus has different applications. To understand Longuenesse’s ideas about the schema, the categories, and the logical forms of Quantity, we have to consider a few of these applications in more detail.
One other representation we can apply quantitative synthesis to, is the pure intuition ofspace. Quantitative synthesis enables us to representspatial
manifoldsas manifolds. What is a spatial manifold? Consider, for instance, a line. A line is a pure spatial shape. Just like the empirical intuition of a house, this line contains a “manifold” of representations. It contains a “manifold” of smaller lines. This manifold can only be represented as
manifold if we subject our representation of the line to quantitative synthesis (265-6). Quantitative synthesis enables us to distinguish and combine parts of the line. As we will see later, such a quantitative synthesis is necessary if we want tomeasure the line.
The quantitative synthesis of space has multiple applications. First, we should note that an empirical intuition like the intuition of a house is also a
spatialintuition. The house has a certain spatialshape. When we apprehend a manifold of representations contained in the empirical intuition of a house, we can apprehend a manifold ofsensations: colours, scents, sounds, etc., but
also a manifold ofspatial forms. When we apprehend the various windows of a house, for instance, we apprehend a manifold of spaces contained in the house’s spatial form. We can also consider the manifold of lines of a certain length that are contained in an imaginary line that we can draw between the top of house and the ground. This, we will see, enables us to measure the house.25 The quantitative synthesis of spatial representations has a second application. Geometry alsoconsists in the application of quantitative synthesis on the pure intuition of space (283). How Longuenesse sees this, I will not explain here.
Besides sensations and the pure intuition of space, it seems, quantitative synthesis can synthesize a third kind of sensible representations. Quantita- tive synthesis can be applied to representations that themselvesresult from figurative synthesis.26 This allows us, for instance, to synthesize collections of empirical objects. Once sensible representations are combined into an em- pirical object, this empirical object becomes a singular representation itself. By applying quantitative synthesis to empirical objects, we can represent manifolds of objects as manifolds. This enables us to represents collections of empirical objectsas collections.
Why would this be necessary? Assume, for instance, I stand in front of a table on which are lying twenty apples. Seeing this table at one glance will make me represent the collection of apples, but I will not represent it as collection. I will, for instance, not be able to tell how many apples there are. To represent the collection of applesas collection, I have to apply quantitative synthesis to them. Just like the manifold of representations contained in an intuition of a house cannot, by that intuition, be given to me “as such”, an intuition of a set of apples on a table cannot be given “as such” by one intuition either.
Longuenesse’s ideas about quantitative synthesis can be summarized as follows: First, quantitative synthesis has a role to play within the complex act of figurative synthesis. Combined with qualitative synthesis, it enables us to represents manifolds of empirical representations, which makes these representations available for relational synthesis. Quantitative synthesis, 25SeeKCJ: 271-4. Longuenesse is not very clear about the exact role she assigns to
the quantitative synthesis of space, and how it is related to the quantitative synthesis of sensations (to represent a manifold of empirical representations, both seem to be required). At some points, she suggests the synthesis of space makes the synthesis of empirical manifolds possible (see, for instance,KCJ: p. 38 and 270-2). How she sees this, however, does not become clear. As this question is not too important for the following, I will not attempt to clarify this.
26
Longuenesse does not explicitly say this, but this follows from her ideas about the various roles quantitative synthesis plays. SeeKCJ: chapter 9.
however, also plays certain autonomous roles. Applied to space, quanti- tative synthesis enables us to represent as manifolds, manifolds of spatial shapes. Quantitative synthesis can also be applied to representations that
resultfrom figurative synthesis, like empirical objects. This allows us to rep- resent as manifolds, manifolds of such representations. These autonomous roles of quantitative synthesis, we will see, are important if we want to un- derstand the relationship between quantitative synthesis and the schema, the categories and the logical forms of Quantity.