I would like to explore some thoughts on the synthetic a priori that I have been having as of late. But before we dive into this, let us first make sure that we understand exactly what synthetic a priori propositions are supposed to be. This is important not only for the purposes of this article, but also because they play such an important role in metaphysics more generally.
The idea comes from the work of Immanuel Kant. To wit, he made two distinctions between a priori vs. a posteriori knowledge, on the one hand, and analytic vs. synthetic judgments, on the other. Simply put, a priori knowledge is any knowledge derived independently of experience. Prime examples of this kind of knowledge would be mathematics and logic. A posteriori knowledge is that which is gained through experience. Obvious examples of this come from the natural sciences.
Analytic judgments are propositions in which the predicate is conceptually contained in the subject. One example of this would be "All 3-dimensional bodies occupy space". If we understand the terms used, then we can see at once that the predicate "occupies space" is part of the meaning of the term "3-dimensional body". It follows at once that in an analytic judgment the predicate does not add any new information. Synthetic judgments are propositions in which the predicate is not conceptually contained in the subject. An example of this would be "All tigers are located on earth." We can see that in a synthetic judgment the predicate does indeed add new information.
Having this in mind, the question of how these two dichotomies are related naturally suggests itself. It seems clear that there are analytic judgments that are known a priori, purely conceptual propositions about the meanings of terms provide an obvious example. Too, it is also equally apparent that there are synthetic judgments that are known a posteriori, with the empirical propositions of the natural sciences being examples of these. But can there be synthetic judgments that are known a priori? That is to say, can there be judgments in which the predicate adds new information, but which can be known independently of experience? As Kant first adumbrated, this question is really the question concerning the possibility of metaphysics in general, since metaphysics proposes to be a purely a priori discipline that provides us with new information about about ultimate reality. (Of course, by 'metaphysics' here I mean metaphysics as first philosophy, and not the new naturalistic metaphysics that is now in vogue).
Now I am not interested for the moment in answering this particular question, so I will just take it as a given that there can be such propositions. What I am interested in exploring is, given that we do have such propositions, what are the various possible grounds for coming to acquire them? In what follows I will attempt to categorize the different possible ways of obtaining this knowledge.
To begin we should note that empirical investigation does not provide a sufficient ground for synthetic a priori propositions, for empirical investigation can only ever provide us with a posteriori knowledge. So too, the characterization postulate does not work either, for this only ever provides us with analytic a priori judgments about the nature of objects. But one obvious way is that which Kant himself provided; namely the transcendental intuitions of sensibility and the categories of understanding. In this way, synthetic a priori judgments are grounded in the very structure of the human mind.
I think another possible ground is the cartesian doctrine of clear and distinct ideas. To wit, we can gain access to synthetic a priori truths through an intuitive grasp of their content; the idea being that we can tell immediately, using nondiscursive methods, that certain synthetic propositions are apodeictic, thus delineating them as synthetic a priori truths. We can appeal to the example of intuitionism in ethics here.
Perhaps another way is the doctrine of mimesis, familiar from Plato's work. Under this doctrine, we first gain knowledge of synthetic a priori truths prior to our births by means of some form of sensuous experience. After our births we retain some faint memory of these experiences, and these can be uncovered through various means (whether that be mystical, rational, or otherwise).
Finally, there is divine revelation. Under this model, a deity or group of deities uses some means or other to directly inform us of synthetic a priori truths, and the very quality of such revelations provides epistemological assurance of their truth. Such revelations can come in a variety of forms, with scriptural inclusion and theophany being obvious examples. The divine revelation approach is also of interest because it provides some very intriguing connections between philosophy and theology.
That's all I have for this post. I just find this a perennially interesting topic, and I wanted to be sure to record my current thinking on the matter. Please be sure to let me know if I have missed out on any other possible methods.
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