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I don't understand the difference between Schema and Synthesis. I have read the First Critique and I see Kant is using the words "schema" and "synthesis" as "intermediate" between intuitions and understanding in different places.

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(Disclaimer: I am aware of the fact that there still is a lot of debate going on regarding what the schemata really are and whether or to what extent the argument works. This is a very basic question, hence a very basic answer)

The answer can be found in A137-38|B176-77. I will quote first and explain later.

[P]ure concepts of the understanding, however, in comparison with empirical (indeed in general sensible) intuitions, are entirely un- homogeneous, and can never be encountered in any intuition. Now how is the subsumption of the latter under the former, thus the application of the category to appearances possible, since no one would say that the category, eg., causality, could also be intuited through the senses and is contained in the appearance? This question, so natural and important, is really the cause which makes a transcendental doctrine of the power of judgment necessary, in order, namely, to show the possibility of applying pure concepts of the understanding to appearances in general. In all other sciences, where the concepts through which the object is thought in general are not so different and heterogeneous from those that represent it in concreto, as it is given, it is unnecessary to offer a special discussion of the application of the former to the latter.

Now it is clear that there must be a third thing, which must stand in homogeneity with the category on the one hand and the appearance on the other, and makes possible the application of the former to the lat- ter, This mediating representation must be pure (without anything empirical) and yet intellectual on the one hand and sensible on the other. Such a representation is the transcendental schema.

Effectively, he makes a logical point here: If there are two things that are inhomogenious in nature (like empirical, sensual intuitions and pure concepts) and these things are thought to be synthesised into a single representation of an object (via empirical synthesis), they need something connecting between them, a copula that has aspects of both fundamentally different things and makes a coming together possible (Note: 'Copula' as technical term can mean something entirely different depending on the context. I am using it as "the form of the connection between the elements of a judgement" here). That's what schemata are, the "in-between" that allows to bring categorically different things together.

In the quote, this is the third he is speaking of. Similar remarks can be found e.g. in the Blomberg Logic, Ak. 9:207, the Hechsel Logic 9:88-89.

In other words: Synthesis is the act of bringing intuitions and pure concepts together, while schemata are the common denominator that allows for bringing them together in the first place, hence an a priori condition of synthesis.

Possible sources of confusion

A confusing aspect of this can be that both the schemata themselves and the outcomes of synthesis - i.e. synthetic representations of determinate objects - can be characterised as "intermediates" between intuition and pure concept. One difference here is that between before and after the fact (the act of synthesis), others are where in the process of empirical synthesis (apprehension, reproduction, recognition, A97-A105) they are coming into play.

Also, there is a lot of (partially misleading, partially plain wrong) literature reading Kant from Hegel that (i.e. the literature, not Hegel himself!) takes the triad "thesis-antithesis-synthesis" to be the basic form of philosophy for both Kant and Hegel. There, synthesis is often taken to be identical to the Hegelian sublation, hence the (speculatively achieved, spontaneous) "intermediate" itself and not the process of achieving it, which as we have seen is in need of a copula to be possible at all. There, synthesis and schemata are muddled. But this goes beyond the scope of the question.

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