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.
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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?
We already mentioned some relevant factors. Kant makes geometrical knowledge in a sense subjective. That’s a major disappointment, one might say. Most philosophers had certainly hoped to be able to defend a much grander claim. Kant “solves” the rationalism-empiricism problem only by as it were belittling geometrical knowledge, which is a very high price to pay.
The main alternative, as we have seen, was to give God a major role in epistemology. So there’s a trade-off: either you pin geometry to God and you can have it be the most amazing thing, the most perfect knowledge, or else you detach it from God and make it stand on its own legs, but then it’s a lot weaker; it’s a mere subjective human thing and no longer this almighty pinnacle of pure intellect.
The exchange rate, as it were, between these two options fluctuated over time. As God became less popular, the cost of switching to Kantianism went down.
But there’s another reason too why Kant’s theory made more sense in the 18th century than in the 17th. Namely what we said before about how Newton’s science was a blow to rationalism.
We spoke about how that was the case. Rationalism requires knowledge to be generated from within the mind. All knowledge needs to be gradually built up from the most simple intuitions, according to the rationalist point of view. In geometry, that meant ruler and compass and other tools for generating geometrical objects. In physics, it meant contact mechanics; that is to say, seeing complex physical phenomena as an aggregate of lots and lots of little collisions of bodies.
Newton’s physics cannot be reduced to contact mechanics. Or to any other simple intuition. It is in fact counterintuitive. So it cannot be generated from within the mind, through an elaboration in thought of the most undoubtable truths. This is why Newtonian physics is a problem for rationalism.
But the story is a bit more general than that. In fact, Newton’s physics can be seen as a blow to philosophy altogether.
From the rationalist point of view, philosophy comes before science. You start with general philosophical thought. “I think therefore I am”: That’s a very general philosophical truth, and you start there because it’s the most knowable. You start by asking yourself what kinds of things are knowable. From that starting point you arrive at the idea that in physics one of the most primitive knowable things is the contact mechanics of bodies.
From this point of view, philosophy is the boss of science. Philosophy is telling science what to do. Before even starting on science, you have already determined through introspection and meditation what the primitive intuitions of physics are. Any science that follows needs to conform to these predetermined rules that philosophy has established beforehand.
From a rationalist point of view, this makes sense. If knowledge fundamentally comes from within the mind, it makes sense to work from the inside out; to start with the most general philosophical core and then build on that to get to things like physics and other stuff that are more connected to the outside world. That’s a core commitment of the rationalist worldview. This is why it requires philosophy to be prior to science, and the boss of science.
Newton does it the other way around. To him, science is the boss of philosophy. This is a natural consequence of his empiricist, “reading backwards” mindset that we have emphasized before. Thought starts not with inward reflection on our basic intuitions, but in the wild jungle of complex phenomena. Science reasons as it were backwards from there to discover the basics principles, such as axioms of geometry and fundamental laws of physics.
If you continue this process one further step you get to philosophy. Just as the laws of physics are whatever is needed to explain the phenomena, so the principles of philosophy are whatever is needed to make that physics possible. So philosophy is subordinated to science. It doesn’t tell science what to do, but the other way around.
To the rationalists, philosophy set the ground rules that science must obey. To the empiricists, to Newton, philosophy merely describes what assumptions are necessary for science after science has already been established. To the rationalists, philosophy is prescriptive: it gives orders, it says how science has to be. To the empiricists, philosophy is descriptive: it’s an observer, a backseat journalist, that merely says how science is, without having any influence over it.
So we see how the basic outlooks of rationalism and empiricism imply these opposite views of the relation between science and philosophy. And Newton’s physics was extremely successful. So its success lent credibility to the empiricist outlook overall, including the demotion of philosophy.
But in fact this is still not the end of it. There is yet another respect in which Newton’s physics dealt an additional death blow to philosophy. Namely on the issue of absolute versus relative space.
Newton clashed with Descartes and Leibniz on this issue as well. It goes like this. What can we know about the spatial properties of a body, such as its position and velocity?
Descartes and Leibniz were relativists about space. Everything we could ever know about positions and velocities of bodies is relative. That is to say, you can only specify the position or speed of a body by comparing it to another body. The chair is so-and-so far from the table. The train is moving away from the station at such-and-such a speed. You cannot speak of the position of the chair or the speed of the train without comparing it to something. You need to relate it to some reference point.
Descartes and Leibniz insisted on this. Here’s how Descartes puts it: “The names ‘place’ or ‘space’ only designate its size, shape and situation among other bodies.” “So when we say that a thing is in a certain place, we understand only that it is in a certain situation in relation to other things.” Leibniz agreed. “Motion is nothing but a change in the positions of bodies with respect to one another, and so, motion is not something absolute, but consists in a relation.”
It takes two to tango, and it takes two bodies to be able to speak of position and velocity. Because you can only describe the position or velocity of the second body by using the first as a reference point.
If there was only one body in the universe, it wouldn’t make any sense to ask whether it was moving or not. Since there’s nothing to use as a reference point, the very concept of motion becomes meaningless is such a situation. According to the relativist conception of space.
This fits very well with our previous emphasis on operations in geometry. Relative positions and relative velocities correspond very well to operations. You can specify what it means for one object to be so-and-so far from another object, or moving with such-and-such a speed with respect to the other object, in terms of concrete measurements. I take a measuring tape, I stretch it from one to the other, that’s how far apart they are.
If there is only one object in the universe, there is no operation we can perform to check whether it is moving or not. So to introduce the idea of every body having some absolute state of motion, independently of any other body, is equivalent to introducing concepts by means other than operations. We know from geometry that this is dangerous, as we saw with the superright triangle and other examples.
Yet Newton does exactly this. Newtonian physics presupposes absolute space. That is to say, it assumes that every body has some definitive position and velocity, completely independently of any other body, and completely independently of what is measurable or knowable to us.
So from the Newtonian, absolutist point of view, if there is only a single object in the universe, then that object still has some definite velocity. It’s either moving or not. Whether it’s moving or not is physically undetectable. There is no way to tell, with a physical experiment, whether it is moving or not. Nevertheless, the question of whether it is “really” moving or not still makes sense and has a definite answer, according to Newton.
This notion–that any body has an “absolute” position and velocity–is necessary for Newton’s physics. Think of the law of inertia. It says: If there is no outside force acting on a body, then the body keeps going in a straight line with the same speed. Forever. Like a metal ball rolling on a marble table, when there is no friction and no obstacles, it keeps going with the same velocity. Without external influence, the state of motion remains the same.
But note that this law talks about the state of motion of a body without reference to other bodies. The law of inertia presupposes that the body has some inherent velocity, a true velocity. That’s the thing that stays the same in absence of interference. Obviously this is not dependent on some particular reference point. The body in and of itself has a state of motion associated with it. The state of motion of the body is an absolute property, not a relative one.
This clash between the absolute and relative space points of view is another clash between science and philosophy. Relative space is clearly the “best” view in terms of philosophy. The philosophical objections to absolute space are very compelling: Absolute space is unknowable. Absolute space introduces concepts that are empirically untestable, unverifiable, unoperationalisable.
The reply from the other side, from Newton’s side, is not to dispute that philosophy is on the side of relative space. Instead it is to belittle the authority of philosophical arguments. Indeed, absolute space makes no sense philosophically. But the conclusion from this is: tough break for philosophy.
Absolute space is a necessary precondition to state the law of inertia, and the law of inertia is an integral part of Newton’s extremely powerful physics, so inertia and hence absolute space must be accepted. Philosophy is just going to have to deal with it.
So this once again reinforces Newton’s point that philosophy is basically a spectator sport. Philosophy can’t tell science what to do. If philosophy clashes with science, as it does regarding absolute space, then philosophy has to give way.
Physicist Stephen Hawking famously declared that “philosophy is dead.” He had in mind 20th-century developments. That’s how many modern scientists think. But philosophy was dead once before. Newton killed philosophy.
If you want to get somewhere in science and mathematics, you can’t get caught up in pointless speculations and debates about “what it all means.” You just have to do the math, get on with it. That was the case in the 18th century, and again in the 20th century.
Another prominent modern physicist, Lee Smolin, put it as follows: “When I learned physics in the 1970s, it was almost as if we were being taught to look down on people who thought about foundational problems. When we asked about the foundational issues in quantum theory, we were told that no one fully understood them but that concern with them was no longer part of science. The job was to take quantum mechanics as given and apply it to new problems. The spirit was pragmatic; ‘Shut up and calculate’ was the mantra. People who couldn’t let go of their misgivings over the meaning of quantum theory were regarded as losers who couldn’t do the work.”
It was exactly the same thing in the 18th century. Then too scientists and mathematicians figured they were better off just ignoring philosophy. And with good reason since Newton’s physics was an obvious winner in terms of mathematics and science, but a complete non-starter philosophically according to many.
The greatest mathematician and physicist of the 18th century, Euler, realized this perfectly well. He knew that absolute space was junk philosophy but essential to science.
He knew that the law of inertia demanded absolute space. As Euler says: “For if space and place were nothing but the relation among co-existing bodies, what would be the same direction? Identity of direction, which is an essential circumstance in the general principles of motion, is not to be explicated by the relation of co-existing bodies.”
Euler also knew that there were powerful philosophical objections to absolute space. The objections of Descartes and Leibniz that I already mentioned. Let me quote here how Ernst Mach later made the same point in the late 19th century. Mach is basically reviving the 17th-century criticism of absolute space. Here’s how Mach puts it:
“Absolute space and absolute motion are pure things of thought, pure mental constructs, that cannot be produced in experience. [They have] therefore neither a practical nor a scientific value; and no one is justified in saying that he knows aught about it. It is an idle metaphysical conception. All our principles of mechanics are experimental knowledge concerning the relative positions and motions of bodies. No one is warranted in extending these principles beyond the boundaries of experience. In fact, such an extension is meaningless, as no one [can] make [any] use of it.”
Euler and others in the 18th century were aware of this problem with the notion of absolute space that is so essential to Newtonian science. They didn’t know how to solve this philosophical problem, except to ignore philosophy altogether. Euler pretty much says so. Listen to this quote:
“I do not want to enter the discussion of the objections that are made against the reality of [absolute] space and place; since having demonstrated that this reality can no longer be drawn into doubt, it follows necessarily that all these objections must be poorly founded; even if we were not in a position to respond to them.”
So Euler admits that he cannot answer the philosophical objections. Instead his solution is: forget philosophy. Philosophy became obsolete with the Newtonian revolution in science. It was out of touch.
Kant is the savior of philosophy. Kant makes philosophy relevant to science again, after a century of being obsolete. Kant’s theory is a way to bring philosophy up to date with science. It is a philosophy that is compatible with Newtonian science, unlike earlier versions of rationalism.
Against this background we can understand why Kant was willing to make mathematical knowledge subjective. That part of his theory was a huge betrayal of a major tenet of classical rationalism. But times had become desperate enough. Philosophy was the laughing stock of scientists. It had to do something, anything.
So Kant decided to bite the bullet on subjectivity in order to at least salvage something of philosophy. Save what can be saved.
Rationalism had once been a mighty kingdom, but it was bleeding territory. Newton’s science was taking the world by storm, and it seemed a real risk that rationalism would not only lose ground but might even be wiped off the map altogether.
Kant’s plan for saving rationalism shows how far it had fallen. In its glory days, rationalism would have scoffed at the notion that geometry is subjective. But now, it was that or death. Like royalty eating peasants’ porridge, rationalism had to adapt or die. Rationalism had to sacrifice the pride of its forefathers–the objective truth of geometry.
But despite this humiliating concession, Kant’s reinvention of rationalism was an astonishing success. Rationalism was back with a vengeance.
Not only was rationalism no longer obsolete or out of touch with science, it was even ahead of the game. Kant had not only stopped the rot but even brought rationalism back on the winning side. Kant’s account not only showed that some parts of classical rationalism could be saved; it also provided the best available account of how the success of Newtonian science could be explained philosophically. Where people like Euler had merely given up on philosophy because of the magnitude of the problems it faced, Kant had shown that philosophy could answer the challenge and more. Philosophy was relevant again. Philosophy was no longer dead.