Summary: Critique of Pure Reason (page 6)

In reality, the mathematician reasons not about the empirical figure of a circle but about the a priori concept of the circle and the formal laws governing mathematical figures: he discovered that he should not rely on what he observed in the figure, nor even on the mere concept of it, […] but that he had to construct this figure himself, through what he conceived and represented a priori, according to concepts¹.

With mathematics, as with logic, we see the emergence of a new kind of knowledge.

We naturally tend to assume that all knowledge is empirical—that is, that learning requires experience of the external world.

However, logic and mathematics show us that another kind of knowledge exists: a priori knowledge. This knowledge does not concern the external world but rather the a priori concepts and laws of our understanding.

But what about physics? Physics seems to be based on experience of the external world. Is it not the empirical science par excellence?

In fact, it too relies—at least in part—on a priori concepts.

Physicists, far from passively observing whatever nature happens to reveal to them, organise experiments. Kant refers to the famous example of Galileo, who made balls roll down an inclined plane with a gravitational force he himself determined. He also mentions the experiments of Stahl and Torricelli.

This led to a profound realisation among physicists:

They understood that reason perceives only what it produces itself according to its own design, that it must take the lead with principles governing its judgements according to constant laws, and that it must compel nature to answer its questions rather than simply allowing itself to be led by nature, as if on a leash.

In other words, physicists came to appreciate the value of a priori knowledge: the goal is not to abandon experience but to use the a priori concepts and laws of the understanding to structure experience and derive useful results:

Reason must address nature while holding, in one hand, its principles […] and, in the other, the experiment it has devised according to these principles—not merely to receive nature’s teachings in the manner of a pupil who accepts whatever the teacher says, but as a judge exercising authority, compelling witnesses to answer the questions put to them.

This is what allowed physics to become a science—whereas before, it had been nothing more than an empirical groping in the dark.

This passage helps clarify an apparent contradiction: why does Kant reject metaphysics on the grounds that it relies solely on a priori concepts, while at the same time considering the results of mathematics and logic—which also rely on a priori concepts—as necessary and true laws?

Why does he state in the Introduction that universal laws are found in the a priori, while in the opening of the Preface the a priori is associated with metaphysics and illusory knowledge?

Why does Kant recognise the validity of a priori knowledge?

We will later examine the answer to this fundamental problem.

For now, we must continue our reading in order to delve deeper into Kantian thought.

At this stage, we should retain the idea that for Kant, there exist valid forms of a priori knowledge, such as mathematics, logic, and physics. One could say that the scientific nature of a discipline resides precisely in the extent to which the a priori governs it. Experience alone is insufficient to establish its scientific character.

Why? Because, as we saw in the Introduction, only a priori judgements are necessary and universal. Empirical judgements—such as swans are white—can always be overturned by a new experience that contradicts previous observations (such as the discovery of a black swan).

¹ Our translation. The references for the quotations are available in the book Kant: A Close Reading