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On Kant’s Synthetic A Priori Propositions

Kant showed that synthetic a priori judgements were possible. Here’s how. Quotes taken from Prolegomena to Any Future Metaphysics.

The 18th century saw the rise of the British Empiricists, and philosophy was dominated by the figures of Locke, Berkeley, and Hume. All had a theory of knowledge that presumed knowledge itself to be only what is conversant in the mind: Locke had a type of inconsistency that allowed for three types of knowledge despite this, Berkeley abolished all matter, stating there were only minds, and Hume maintained all theoretic consistency which led him to solipsism. All fell to subjectivism.

But, of course, all of these men were social cosmopolitans — none of which conformed their practical life to their philosophy.

Yet, this was all in a sense the background to which the founder of German Idealism, Immanuel Kant (1724–1804), reacted — stating even Hume awoke him from his “dogmatic slumber.”

Kant was a disciplined person, raised on piety but wholly liberal. He had regular habits that people could set their watches to. He wrote vigorously. His endeavors were usually scientific, as Bertrand Russell recalls,

“After the earthquake of Lisbon, he wrote on the theory of earthquakes; he wrote a treatise on wind, and a short essay on whether the west wind in Europe was moist because it had crossed the Atlantic Ocean. Physical geography was a subject in which he took great interest.”

Yet Immanuel Kant is truly regarded as a guidepost to major shifts in understanding metaphysics, morality, epistemology, and aesthetics.

Although his most important work is the Critique of Pure Reason, a lesser known essay entitled Prolegomena to Any Future Metaphysics offers a particular objection not as explicitly found anywhere in the rest of his work; namely, an objection to his predecessor’s understanding of pure mathematics and a priori truths.

It might seem far off, but this objection actually had fundamental importance regarding the conception of knowledge and how we might attain it. First, let’s look at some of the types of knowledge propositions Kant distinguished between.

Types of knowledge propositions

Kant makes two distinctions — both in the Critique and in the Prolegomena— bewteen types of propositions. Propositions that are analytic, as opposed to synthetic, and propositions that are a priori, as opposed to empirical or a posteriori.

An anaylitic proposition is a type of logical proposition that exists such that the predicate is contained within the subject so that the notion of the subject denotes the predicate. For example, all green-eyed women are women, or all bachelors are unmarried. You only need to know the truth of the constituent terms in both of these propositons to be aware of the truth of the proposition as a whole.

These propositions depend “wholly on the Principle of Contradiction”, according to Kant. That is, the contradiction of an analytic proposition is itself self-contradictory (same is true of the contrapositive). And it is for this “reason that all analytic judgements are a priori judgments”; that is, judgements known independent of experience.

A synthetic proposition is one which requires knowledge of the world which appears to us, as it were, non-analytically.

Synthetic claims require a “synthesis” of the subject and empirical evidence (additional concepts) such that new claims are made with the conjunction of the subject and the new concepts. These propositions do not rely on the Principle of Contradiction, but must be grounded elsewhere. It is worth noting that the synthetic proposition is, in fact, yet subject to the Principle of Contradiction; however, the entirety of the contents of the proposition can not be deduced from it.

In other words, synthetic truths require reference or knowledge of outside objects; i.e., we know synthetic propositions through experience.

For example, consider the propositon “Alexander was a remarkable king.” I can not, by merely knowing the truth of the constituent terms, discern the truth of the propositon as a whole. I need to first know what Alexander accomplished, his many actions, reports of his campaigns, etc.

This is why philosophy up until Kant decided all synthetic propositions are only known by experience; that is they are necessarily empirical truths. However, Kant rejected this claim and showed that a priori synthetic judgements were possible.

How he showed this we will turn to presently, considering his use of pure mathematics in his Prolegomena mentioned earlier.

Kant’s Objection to his Predecessors: A priori synthetic truths

Kant’s objection can be stated as the following: the previous two intellectual traditions did not sufficiently explain or distinguish between analytic and a priori and synthetic and a posteriori such that all mathematical (and a priori) truths (as claimed Descartes, Locke, Leibniz, and Hume) relied on the principle of “contradiction alone.”

Moreover, a priori synthetic truths are possible.

For the entirety of the modern period the Rationalists and the Empiricists accepted the breakdown we listed above: a priori truths were analytic, empirical truths were synthetic.

More to the point, most all Modern thinkers were, at least to some degree mathematical platonists (i.e., the concept of the number “two” has its derivation from a pure form and can be known analytically and a priori). Kant, however, through claiming that a second type of synthetic judgement existed — mathematical judgements — offered the view that synthetic claims can, in fact, be known a priori.

This is something no Rationalist had committed herself to, and something the Empiricists, prominently Hume, considered exhausted and untenable.

Kant famously provided the mathematical example.

He writes that it first might be thought that “the proposition 7 + 5 = 12 contains merely their union” and thus the sum and the concept of 12. However, Kant goes on to say: “the concept of 12 is by no means thought by merely thinking of the combination of seven and five; analyze this sum as we may, we shall not discover twelve in the concept.”

That is, although we may realize the arithmetic axiom of summation, the concept of “7”, and the concept of “5”, it is not the case that we have any knowledge of the “number that unites them”.

Instead, it requires the understanding of both the concept “7” and “5”, and the addition of the successive units of one to the other. “Hence”, Kant claims, “our concept . . . is really amplified [in that] we add to the first concept a second one not thought in it.”

In a word, the proposition “7 + 5 = 12” is synthetic in that the proposition does not contain a predicate that is contained in the subject, and a priori in that the origin of knowledge of the individual concepts are not empirical, but can be known merely from analysis.

So, the non-empirical concepts (in this case, numbers), in combination with intuition superimposed atop their understanding, yield to form a syntheses of their relation into a separate concept from their constitutive logical parts (7 and 5 construct with intuition to make 12).

This is a remarkably swift and beautiful argument.

Again, this, for Kant, was proof that there could be synthetic a priori claims through mathematics — a major objection to all previous arguments that maintained mathematics’ analyticity. Hume and others before him seem to have allowed this fact, says Kant, to “altogether escape” their observation and enquiry.

It is worth noting that Kant identifies a feature that distinguishes pure mathematical cognition among all other a priori cognition: its inability to proceed from concepts; i.e., only “by means of the construction of concepts.”

This intuitive syntheticity contrived by Kant’s example also served as an objection to the idea that the reliance was wholly on the Law of Contradiction in mathematics. However, Kant applied this argument to other realms, claiming to find other synthetic a priori propositions in Euclidean geometry and logic, for example.

Geometric principles, such as those seen in synthetic relational concepts fit this category. Kant considers the synthetic concept of two points and it’s having the shortest distance between the two in that of a straight line going through them. The syntheses of the concept of a straight line and shortness is necessary in understanding, as “my concept of straight has nothing to do with quantity”, but only quality.

That is, the concept of “shortness” is not contained as a predicate in the concept of “straightness” or a “straight line”; therefore, it requires intuition to synthesize the two concepts and arrive at the proposition that the shortest distance between two points is a straight line between them.

Therefore, we can conclude that for Kant, “all principles of geometry are no less synthetic” than those of pure mathematics; viz., the objection holds its force in geometry as in mathematics.

What did not make it into Kant’s perview here are tautologies and other basic logical truths. As far as basic logical truths are concerned (a=a, ~(P ^ ~P), modus ponens, modus tollens, etc.), they seem to escape Kant’s criticism and fall outside the scope of his argument by way of their being analytic; that is, one can know the principle of identity by knowing the thing identified; the proposition A = A can be known from merely understanding the concept, A — there is no synthesis required.

Moreover, these principles themselves rely wholly upon the Principle of Contradiction, a sufficient condition for them being analytic in Kant’s view; the principles are valid from “mere concepts.” In another way, the basic logical truths contain a “duplicity of expression”; e.g., the concept of A being itself is contained in the concept of A. The logical truth, then, can be said to be unaffected by Kant’s introduction of intuitive syntheticity.

Brief Afterward

In all, Kant diverged from his contemporary tradition and introduced an entirely new concept of synthetic a priori propositions into the dialectic. His Prolegomena (along with his Critique) has permanently shifted our understanding of synthetic propositions and began the ongoing debate surrounding their potential a priori grounds for knowing them.

Kant’s successors, including Fichte and Schelling, maintained the core of his Idealism, including much of his theories on knowledge and a priority and are perhaps responsible for his being considered a Romantic. However, it was not until Hegel that we saw much of the progress Kant had made room for. His theory of knowledge impacted many later debates, including the phenomenological one regarding the structure and processes of consciousness. This is largely played out by the Existentialists of the twentieth century, including Jean-Paul Sartre.

Indeed Kant, for better or worse, has a position and legacy few thinkers have or will ever have. Much of his thinking has waned in popularity in recent history, but he is still often the benchmark for philosophical discussion. Reading him provokes a scientific and a mystical sensation few in history have commanded. His volumes are anything but terse, and at times require a glossary of terms along with many reads to fully take in.

Still, Kant is a thinker whose innovations corresponded with great philosophical shifts and lasting importance.




Anthony DiMauro is a freelance writer in New York City. J.D. Candidate. NYU philosophy alum. You can follow him on Twitter @AnthonyMDiMauro

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Anthony DiMauro

Anthony DiMauro

Anthony DiMauro is a freelance writer in New York City. J.D. Candidate. NYU philosophy alum. You can follow him on Twitter @AnthonyMDiMauro

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