Synthetic a priori Judgments and Geometry

In the Prolegomena, Kant uses geometry as an example to show that we do have synthetic a priori knowledge. He argues that grasping how this kind of knowledge is possible there can lead to insights concerning the possibility (and the limits) of synthetic a priori knowledge in other areas, such as metaphysics. We will focus here on certain aspects of what he means in calling some geometrical knowledge synthetic and in calling it a priori. This, together with the example of geometrical thinking linked to below, should help us to see why he thinks that experience–or knowledge of objects in the world–presupposes both sensory input and synthetic a priori principles that allow us to ‘see’, ‘read’, and ‘interpret’ this input in determinate ways.


Reproduction of a proof from Euclid’s Elements (Public Domain: via Wikipedia Commons)


Some geometrical knowledge is synthetic, meaning that it involves judgments that go beyond merely analyzing concepts and making more explicit what is already thought implicitly within these concepts (e.g., analyzing the concept ‘rectangle’ and making it explicit that anything that qualifies as a rectangle has to have angles, and that these angles have to be right angles–those features of rectangles are already thought in the concept ‘rectangle’).

Synthetic knowledge involving things like points, lines, planes, circles, triangles, rectangles, etc. involves going beyond mere concepts, and providing imaginary correlates for these concepts that enable us to ‘draw’ individuals that stand in for the general kind of figure under consideration. Kant calls this process construction in intuition. After we proceed synthetically in constructing a triangle in a particular way, we can then analyze the results of this construction to determine that the triangle has certain features (e.g., sides of identical length) that are not included in the thought ofwhat it is to be a triangle.

A priori

This kind of knowledge, despite requiring that we go beyond mere concepts, is also a priori –it does not depend on actual experience of any actual triangle in nature; it depends on thought and construction in the imagination (even if we also draw on paper to help us keep track of what’s going on in the imagination). Unlike empirical knowledge of individual bodies in nature, the results we obtain from the construction of a single geometrical figure (in accordance with definitions, axioms, and a plan) are perfectly general. We can know that these same results will obtain for any and all figures that are constructed in the same way.

One thing that Hume has neglected, if Kant is right, is that geometrical knowledge is not simply a matter of reasoning about relations of ideas. Instead, it is a matter of reasoning about the (actual and possible) external forms and external relations of (actually and possibly) existing things by way of an intellectual faculty of concepts (understanding) working together with a sensible faculty of intuitions (imagination).

Of course, empirical knowledge of what particular causes in nature have what particular effects does not proceed in exactly the same way as a priori knowledge in geometry. Kant insists, however, that there are some extremely important similarities between the way the mind works in constructing geometrical figures and the way it works in constructing objects of external sense experience. The same spatial forms that we treat without any empirical matter in geometry, as well as some logical forms that we will discuss next time, give both intuitive and conceptual structure to the matter of sense experience, i.e., sensations.

Before Thursday’s class, I would like you to read this post on Descartes and Geometrical Demonstration to get a sense of the kind of constructive process that both Descartes and Kant are referring to when they discuss distinctions between synthetic methods and analytic methods. Share your comments and any questions you have on the Discussion board in Angel titled ’Synthetic a priori Judgments and Geometry’.

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