Kant developed a philosophy of geometry that explained how geometry can be both knowable in pure thought and applicable to physical reality. Namely, because geometry is built into not only our minds but also the way in which we perceive the world. In this way, Kant solved the applicability problem of classical rationalism, albeit at the cost of making our perception of the world around us inextricably subjective. Kant’s theory also showed how rationalism, and philosophy generally, could be reconciled with Newtonian science, with which it had been seen as embarrassingly out of touch. In particular, Kant’s perspective shows how Newton’s notion of absolute space, which had seemed philosophically repugnant, can be accommodated from an epistemological point of view.

Transcript

Rationalism says that geometrical knowledge comes from pure thought. Empiricism says that it comes from sensory experience. Neither is very satisfactory, because geometry is clearly both: it’s too good a fit for the physical world to be only thought, and it relies too much on abstract proofs to be only experience.

So it seems the “right” philosophy of mathematics must be a little bit of both. But how? Rationalism and empiricism mix like oil and water. They are so different, so opposite, that it seems impossible to find any sort of middle ground that has half of each.

Nevertheless there is a golden mean of sorts. The philosophy of Kant. Immanuel Kant, the late 18th-century philosopher. His view of geometry is in a way the best of both worlds: combining the best rationalism with the best of empiricism. Let’s see how he pulled that off.

Kant’s theory is another innateness theory. Geometry is innate; it is hardwired into our minds. That’s what the rationalists said too, of course. Innateness was why the uneducated boy in Plato’s Meno could reach substantial geometrical insights without instruction. The innate intuitions of geometry are reliable because God is not a deceiver, said Descartes. And they apply to the physical world, because the Creator put into our minds the same ideas he had used to design the universe, said Kepler and others.

But Kant’s innateness is different. He lived a century and a half after Kepler and Descartes, and the reliance on God to support the rationalist worldview had become much less fashionable during the intervening years. And indeed Kant does away with it.

By taking geometry to be innate, Kant automatically inherits the best of rationalism. That is to say, he is able to account for the prominent role of pure reason in geometry in a convincing way, just as the earlier rationalists had done before him.

But Kant solves the weakness of rationalism very differently. If geometry is innate and susceptible to purely theoretical elaboration, then why does it always agree so well with experience? Not because “God was a geometer,” as Plato and Kepler said. No, Kant’s solution is: Not only are geometrical principles innate in our thoughts, they are also innate in our perceptions.

Aha, plot twist! The so-called success of mathematics in the real world is no miracle. It’s a rigged game. Our eyes and our senses are biased. They can only see Euclidean geometry.

It’s an illusion to think that we can compare our mathematical deductions with reality. We think we can “test” whether theoretically established theorems are true or not in the physical world; for example, by measuring the sides of right-angle triangles and comparing the results to what Euclid says it should be. But we were naive to think that these two things were really independent. Just as our thoughts are shaped by our innate intuitions, so also our perceptions of physical reality are shaped by the same intuitions.

We don’t have direct access to “raw data” about the physical world. When we think we observe the world, we really observe an interpreted version of the world. The mind can only process information that is interpreted or converted to fit a particular format. In terms of geometry, that means Euclidean geometry.

We see the world through Euclid-colored glasses. Like those pink sunglasses that John Lennon used to wear. Of course if you wear pink glasses, then everything looks pink. But of course John Lennon could not say: aha, I told you, the world is rosy; I argued theoretically that the world should be rosy, and now I have confirmed by observation of the actual world is in fact rosy, because look how everything is pink. John Lennon would be fooling himself if he argued that way.

Euclidean geometry is exactly like that, according to Kant. With experience and observation, we do not discover that Euclidean geometry applies to the real world. What we really observe is our own biases. We discover not facts about the world, but facts about what kind of glasses we are wearing.

John Lennon made the world rosy by putting on his glasses, and in the same way we make the world Euclidean by a hidden lens in the mind’s eye.

Think of a chili pepper. This pepper proves that Kant was right. How so? Think about it. What are the characteristics of a chili pepper?

It’s red. What is red? Is the pepper “really” red? That is to say, is its redness an objective property of reality? Well, yes and no, right? The way science accounts for colors is in terms of wave length of light. What we in our minds regard as red correspond in reality to a particular frequency of light waves.

So there’s something there corresponding to red, but our minds put a major interpretative spin on it. Red is really just a wavelength, “just as number,” so to speak. But our minds turn it into something more qualitative.

And the mind is very selective as well. The remote control of your TV sends its signal using infrared light, which the minds chooses not to see at all. Even though infrared light is exactly the same kind of thing as the color red from the pepper. They are exactly the same kind of waves, only with slightly different frequencies. “Objectively” they are very similar. But subjectively, in our minds, they couldn’t be more different. One is the color red that is one of the fundamental categories in terms of which we navigate reality, and the other is completely invisible.

So perception is very far from direct access to “objective” raw data. The mind interferes very heavily: it selects, transforms, distorts. Maybe when we think Euclidean geometry fits reality it’s only because of this. Only because things that don’t fit a Euclidean mold are as invisible to us as infrared light.

The chili pepper is also very hot to taste. Again, is that an objective property of pepper itself, or is it just a subjective matter of how it interacts with our tongues?

In fact, birds eat chili peppers and to them they are not hot. Chili peppers evolved to be hot because it’s better to be eaten by birds than to be eaten by mammals. The seeds spread better that way. So chili peppers developed this characteristic of being repellently hot to mammals like us, but but perfectly appetizing to birds.

Could it be the same with our geometrical intuitions? “Two lines cannot enclose a space,” Euclid says. The very thought is repellent! Well, maybe that’s just our subjective experience, just as we find it repellent to bite into a hot pepper. Maybe other creatures have completely different geometrical intuitions. Just as birds think pepper are not hot, maybe they also think space has five dimensions or whatever. Who knows.

Different animals can have all kinds of different intuitions and modes of perception that fits the way they navigate the world, the way they find food, and so on. Maybe Euclidean geometry is just something that happened to be convenient to us, just as night vision is convenient to a cat, or a heightened sense of smell is to a dog.

That would make geometrical experience quite subjective. Or at least subjectively contaminated, or entangled with subjective factors.

This is how Kant is able to bridge the gap between rationalism and empiricism. We can sit in a closed room and figure out in our heads geometry that then turns out to be true in the real world. Because, says Kant, what is really happening is that the mind is analyzing itself. What we discover through geometrical meditation is not objective facts, but facts about the geometrical preconceptions hardwired into our minds and the consequences of those conceptions. And these results apply to the world not because they are really out there but because our perception actively processes and converts and interprets any sensory input in Euclidean terms.

As Kant himself says: “Space is not something objective and real; instead, it is subjective and ideal, and originates from the mind’s nature as a scheme, as it were, for coordinating everything sensed externally.” “It is from the human point of view only that we can speak of space, extended objects, etc.”

In the introduction I framed the issue in a way that is flattering to Kant. I said: Everybody knew that you needed some kind of middle ground between rationalism and empiricism, but no one could figure out how to do it. Until Kant, he finally cracked the nut.

But maybe that’s the wrong way to look at it. It often is, this kind of narrative. “Everybody tried to do such-and-such but no one could do it, until finally one guy was smart enough to figure it out.” That’s not usually how history works. If we look into the details we often find that contextual factors explain why key steps happened at certain times.

It’s true that everyone knew that neither rationalism nor empiricism were perfect for hundreds of years. And Kant’s proposal is certainly a very clever way of tackling that problem. So why did it take all the way to the late 18th century before these ideas were proposed?

Rationalism 2.0: Kant’s philosophy of geometry