The Problem With Geometry

Geometry might be innate in the same way as language. There are many languages, each of which is an equally coherent and viable paradigm of thought, and the same can be said for Euclidean and non-Euclidean geometries. As our native language is shaped by experience, so might our “native geometry” be. Yet substantive innate conceptions may be a precondition for any linguistic or spatial thought to be possible at all, as Chomsky said for language and Kant for geometry. Just as language learning requires singling out, from all the sounds in the environment, only the linguistic ones, so Poincaré articulated criteria for what parts of all sensory data should be regarded as pertaining to geometry.

Transcript

The discovery of non-Euclidean geometry in the early 19th century was quite a wake-up call. It showed that everybody had been a bit naive, you might say.

Here’s an analogy for this. Suppose we had all been speaking one language, let’s say English. And we were all convinced that English is the only natural language. In fact, that question didn’t even arise to us; we simply assumed that English and language is the same thing. We even had philosophers “explaining why” English is a priori necessary. These philosophers had “proved,” they thought, that without English the very notion of linguistic communication or thought is impossible.

And then we discovered that there are French speakers and Chinese speakers. Oops. Very embarrassing. English is not necessary after all. It is not innate, it is not synonymous with language itself. For thousands of years we made those embarrassing mistakes because we were not aware of the existence of other languages.
That’s how it was with geometry. What I said about English corresponds to Euclidean geometry. For thousands of years, nobody thought of Euclidean geometry as one kind of geometry. Everybody thought of it as THE geometry. Geometry and Euclidean geometry was the same thing. Just as an isolated linguistic community thinks their language is THE language.

And the philosophers I spoke of, Kant is an example of that. He argued that Euclidean geometry was a necessary precondition for having spatial experience or spatial perception at all, which is like saying that English is necessary for any kind of linguistic expression.

And already long before Kant, many people had been convinced by how intuitively natural and obvious the axioms of Euclidean geometry feel. Descartes for instance and many others made a lot of this fact. Remember how important it was to Descartes that our intuitions were truths implanted by God in our minds.

Well, we all think our native language is intuitive. And we think other people’s languages are not intuitive. This feeling is so strong that we think it must be objective. When we try to learn a foreign language, it feels impossible that anyone could think that was intuitive. And yet they do.

So apparently our intuitions can deceive us. We feel that our native grammar is much more natural than everyone else’s, but that’s a delusion. It felt like an objective fact, but it turned out to be subjective.

Could it be the same with geometry? Could the alleged naturalness and intuitiveness of Euclidean geometry turn out to be just an arbitrary cultural bias, like thinking English feels more natural than French?

So, ouch, we took quite a hit there with the discovery on non-Euclidean geometry. It exposed our insularity. It showed that things we had thought we had proven to be impossible were in fact perfectly possible and every bit as viable as what we had thought was the only way to do geometry.

And yet there is hope as well. The language analogy doesn’t just expose what an embarrassing mistake we made, or how non-Euclidean geometry hit us where it hurts. The language analogy also suggests a way out; a way to rise from the ashes.

We were wrong about the specific claim that Euclidean geometry is innate, because that’s like saying that English is innate. Nevertheless we were right too, one could argue.

Language is impossible without something innate. Everybody learns their native language with incredible fluency. Every child learns it somehow, with hardly any systematic teaching; they just pick it up naturally. And they do so at a very early age, when their general intelligence is still very limited.

Meanwhile, no animal comes anywhere near achieving the same feat. Nor any adult human for that matter. I’m better than a three-year-old child at any intellectual task, except learning French. Somehow the child is super good at that.

So clearly something about language is innate. The ability of language acquisition is innate. Some kind of general principles of language are innate.

Perhaps geometry is like language in this way. We have some innate geometry. Not specific Euclidean propositions or axioms maybe, but some kind of “geometryness” nonetheless. Some sort of more structural or general principles of geometry than specific propositions, just as our innate linguistic ability doesn’t contain anything specific to any one language but instead has to do with “languageness” in general.

In linguistics, this is called “universal grammar.” Every language has it own grammar, of course, but there are some more general or structural principles of language that are the same for all human languages. This common core is the universal grammar.

Here’s an example of such a principle that belongs to universal grammar. Consider the statement “the man is tall.” It is a declaration, an assertion. You could turn it into a question: “Is the man tall?” We turned the assertion into a question by moving the “is” to the front of the sentence. That’s how you make questions from statements: start with assertions—“the man is tall”—and move the “is” to the front—“is the man tall?” So that’s a recipe for making questions.

But consider now the statement “the man who is tall is in the room.” How do you form the corresponding question? There are two “is”’s in the sentence. So the rule “to form a question, move the is to the front” is ambiguous. Which of the is’s should we move?

It could become: “is the man who is tall in the room?” That works. But if we move the other is to the front we get nonsense: “is the man who tall is in the room?” Well, that didn’t work. It didn’t make a question, it just made gibberish. Even though we followed the same rule as before: move the “is” to the front.

If language was nothing but a social construction then it would be perfectly reasonable for children trying to form questions to come up with the gibberish one. If the child simply extracts general rules from a bunch of examples, it would have been perfectly reasonable for the child to have guessed that the general rule is: move the first “is” to the front of the sentence to form a question. Which would lead to the nonsense question “is the man who tall is in the room?”

In fact, however, “children make many mistakes in language learning, but never mistakes such as [this].” Apparently, “the child is employing a ‘structure-dependent rule’” rather than the much simpler rule to put the first “is” in front. Why? “There seems to be no explanation in terms of ‘communicative efficiency’ or similar considerations. It is certainly absurd to argue that children are trained to use the structure-dependent rule, in this case. The only reasonable conclusion is that Universal Grammar contains the principle that all such rules must be structure-dependent. That is, the child’s mind contains the instruction: Construct a structure-dependent rule, ignoring all structure-independent rules.” So although each language has its own grammar, there are some general principles like that that are universal: common to all languages. Those principles are hard-wired into the mind at birth.

This stuff about there being an innate “universal grammar” is the view Chomsky, the leading 20th-century linguist. The example and explanation I just quoted are from his book Reflections on Language. There are many debates about Chomskyan linguistics, but I’m going to assume Chomsky’s point of view for the purposes of this discussion, because its parallels with geometry are very interesting.

You might say that this view of language is Kantian in a way. We saw that Kant put a lot of emphasis on the necessary preconditions for certain kinds of knowledge. Geometry, for example, is not an external, free-standing theory that we can analyze with our general intellectual capacities. Rather, the fundamental concepts of geometry are bound up with the very cognitive structure of our mind itself.

Some things are purely learned through experience and convention, such as how to play chess or how to dance the tango. But some things are not like that. For instance, they way we experience color. The mind is made to see red and blue and to not see infrared and so on. That’s a fixed, domain-specific property of how our mind works. You can neither learn nor unlearn that through general intelligence. Color experience is just one of those basic things hardwired right into the brain.

From the Chomskyan point of view, language is like that as well. Language is not merely a social construct with man-made rules like chess or tango. Nor is it explicable in terms of general intelligence only.

Chess or tango you can learn by general intelligence. That is to say, if you spend enough time looking at people playing chess or dancing, you can eventually figure out what the rules are by general rational thinking such as pattern recognition, and forming preliminary hypotheses about how you think it works and then observing some more to check if you maybe need to revise the hypothesis to take into account some other possible circumstances or cases.

Color experience is not like that. You don’t learn to experience redness by watching other people. It just is. And if you’re not born with it, then you can’t learn it by general intelligence, like you can learn chess.

Language is similar to color and not similar to chess. You don’t learn color perception by watching others and using general intelligence to figure out the patterns and rules. General intelligence is not sufficient to sustain such a thing.

Many people overestimate the potential of general-purpose intelligence. Both Kant and Chomsky agree about this. Remember the tile of Kant’s work: a critique of pure reason. “Pure reason,” or general-purpose intelligence, is not by itself capable of generating human linguistic capacity or geometric experience.

The capacities of our mind depend much more than people realize on domain-specific conceptions. It is obvious that color experience is a hardwired specific domain of our cognitive structure and isn’t merely the outcome of some pattern-recognition process of general-purpose intelligence. But it’s less obvious that geometry is like that, or that language is like that. But Kant and Chomsky maintain that they are. According to them, we underestimate the extent to which basic geometrical and linguistic conceptions are intertwined with the very nature of our mind and our cognitive capacities.

So the wrong way to think about it would be like this. The human brain is a general-purpose thinking machine. Imagine a person in a prehistoric hunter-gatherer society. This person’s general-intelligence mind might think to itself: Well, it’s great that I’m so smart. I can learn many things, like which plants are poisonous; I can figure things out like how to make fire, how to use tools and so on. But gee, wouldn’t it be handy if I could communicate my thoughts to others. Then we could organise collaborations, learn from each other’s experiences, and so on. I know, let me invent language, that will work for this.

From the Chomskyan point of view this story is wrong because it overestimates the general-purpose mind. In fact, note that I described what the pre-linguistic mind was thinking by using language. But I was talking about a hypothetical stage in history in which there was no language. Does it even make sense to imagine such a thing as thought without language? No, according to Chomsky. The very nature of thought itself cannot be separated from language like that.

The story of the hunter-gatherer inventing language is no more plausible than the story that he invented color experience by discovering that certain wavelengths of electromagnetic radiation were associated with grass, others with fruit, and so on.

Instead of thinking of the mind as starting from general-purpose intelligence and then inventing domain-specific things like color and language, we should perhaps think of it exactly the other way around. The mind is made up of the domain-specific skills. Those are the fundamental cognitive starting-points. Insofar as we have any general-purpose intelligence, that comes from piecing together the domain-specific skills. Not the other way around.

From an evolutionary point of view, the human mind perhaps evolved by adding domain-specific modules one by one: first color, then a hundred thousand years later geometry, then a hundred thousand years later language, and so on. We don’t have general-purpose intelligence. We only have the sum of our modular parts. But eventually these modules became so advanced, and combined in such fruitful and powerful ways, that we fool ourselves into thinking that we have general intelligence, “pure reason.” But at bottom our precious “pure reason” actually still depends more than we realize on domain-specific preconceptions hardwired into our cognitive capacities. That’s what Kant said about geometry and that’s what Chomsky said about language.

So in this way we can “save Kant.” The discovery of non-Euclidean geometries was a blow to Kant’s idea of the innateness of geometry. Kant associated the intuitiveness of Euclidean geometry with its innateness. But native languages are intuitive, yet they are not innate. And geometry could be the same, because just as there are many languages there are many geometries. This shows that intuitive and innate is certainly not the same thing, so it calls into question the Kantian story that the mind is constrained by pre-programmed conceptions.

We save Kant with the rebuttal that in fact language too is innate after all. Even though there are many languages that all differ in fundamental respects, nevertheless there is some universal languageness that is common to all and without which language learning would be impossible in the first place.

Same with geometry. Instead of focusing on the differences between Euclidean and non-Euclidean geometries and concluding from this that no one geometry could be a necessity of thought, we should instead focus on the more fundamental and structural preconceptions common to all geometries, without which any kind of geometry would be unthinkable at all.

Or we can put it like this. Thought presupposes language. When you think, you think in terms of words and sentences. Of course thought does not presuppose any specific language. You can think the same thing in English or German. Nevertheless thought does presuppose that you use some language. There is no “pure thought,” or hardly any, that does not involve words.

It’s funny: thought cannot exist without language, yet you can switch the entire language and still have the same thought. So there’s both dependence and independence.

Kant says basically the same thing but for geometry. You can’t have spatial perception or spatial reasoning without geometrical presuppositions. Just as you can’t think without presupposing some language, so you can’t geometrize without presupposing some geometry.

The choice of which language or which geometry you take as the basis for thought is arbitrary. As Kant says, it’s a synthetic a priori, not an analytic a priori. That is to say, it is not logically necessary that we must use Euclidean geometry as the presupposition for all our spatial experience. But it is necessary that we must make some such presupposition.

Remember, as Kant said, we don’t have direct access to objective physical reality. We only know the outside world through perception which is always necessarily interpreted. The presuppositions of that interpretation are arbitrary—in fact, it’s arbitrary in two ways one might say: one good and one bad. It’s arbitrary in a “bad” way in that it is subjective. It lacks objective justification. But it’s also arbitrary in a “good” sense, namely that it doesn’t necessarily matter all that much which interpretation we choose.

Just like language. It is arbitrary that I’m speaking English. There’s no objective or logical reason for why English is any better than any other language. But it’s also arbitrary in that it doesn’t matter. I could have said the same things in some other language. And in fact it’s only because of my choice of some arbitrary language that I am able to say anything at all.

Same with geometry. Our minds think in terms of Euclidean geometry even though that has no absolute logical justification. Yet it would be a mistake to criticize this as arbitrary subjectivity. Because it is only because I have some geometrical preconceptions at all, no matter how subjective, that I am able to reason spatially and have spatial perception and experience in the first place.

The analogy that geometry is like language is suggestive in other respects as well. Here’s one interesting question. When a child is learning their native language by picking up the speech of their parents and their environment, how does the child know which sounds are language and which sounds are other kinds of noises? It’s a pretty difficult problem, isn’t it?

Suppose you had to program a computer to detect and recognize speech. What criteria could you define by which the computer could tell if any given sound is linguistic or not? Words come in many forms: you can scream them, whisper them, sing them. Those are very different as sounds, but somehow you have to be able to tell that they are all words. And you have to be able to tell that other sounds are not linguistic, such as a doorbell, a barking dog, the sizzling of a frying pan, and so on.

You have the same problem in geometry. Among all the sensory impressions we are bombarded with every second, which ones should be regarded as geometrical, and which not? If geometry is like a language, a child must have some criteria by which to answer this. Just like the child somehow picks out linguistic sounds from the environment and lets that shape their native language, so also the child must pick out geometric features of the environment and let that shape their native geometry. This is how their intuitive geometry can become either Euclidean or non-Euclidean depending on the environment, just as their native language can become English or Russian or whatever.

So: What parts of all our sensory impressions have to do with geometry? You must know that first, before you can start thinking about whether those impressions are Euclidean or non-Euclidean.

Poincaré had a very elegant solution to this problem. Here’s his criterion for telling geometry from non-geometry. It goes like this: Among all sensory impressions, those are geometrical that you can cancel through self-motion.

Let me explain what this means by an example. I have a piece of paper. One side is white and the other side is red. I hold the paper up with the white side facing toward you. Then I rotate it so that the red side is facing you. This is a geometrical transformation: it has to do with rotation, with position. You know that it was geometrical because you could walk around and stand on the other side and then you would see the white side of the paper again. So you could cancel the transformation in impressions, you could restore the original sensory impression, through self-motion. By moving yourself. Not by manipulating the environment, but only by moving around in it.

There are many transformation of sensory impressions that are not like that. That are not cancelable or reversible through self-motion. Including other kinds of switches from white to red. Pour a white liquid, like a lemon sports drink, into a glass. And then pour in something very red, like beet juice or some strawberry syrup. The liquid in the glass went from white to red, just like the paper did when I flipped it over.

But the liquid is different, because you can’t cancel it this time by moving around and looking at it from another point of view. This is precisely why it is not geometrical. The paper example should be interpreted in terms of geometry. If someone asks: what happened? Then for the paper example you would give an explanation in geometrical terms: the object rotated 180 degrees. But for the liquid example you would give an explanation in non-geometrical terms: the red liquid “colors over” the white one by some kind of, I don’t know, chemistry somehow; not geometry anyway.

So there you have a very clear criterion for selecting from the environment which things are to be accounted for in terms of geometry and which not. Cancelability through self-motion.

Before a child can tell if their parents speak French or Russian, they must be able to distinguish which sounds are linguistic at all. And before we can tell if the space around us is Euclidean or non-Euclidean, we must first be able to distinguish which sensory impressions have to do with geometry at all. Poincaré’s criterion in terms of self-motion answers this problem.

So this suggests that it is only through motion that we can impose a geometric interpretation on our visual impressions. It may feel to us as if our sense of sight is inherently geometrical: geometry is visual, it lives in the eyes. But Poincaré’s perspective suggests that it’s more complicated than that.

Vision becomes endowed with geometry only through its interaction with self-motion. If we could not move ourselves or our eyes, our sense of sight would be as un-geometrical as our sense of taste or smell. It would be just a bunch of qualitative impressions with no particular structure.

With sense and smell, you can tell when one thing is different from another, but you can’t do much more than that. There is no “Pythagorean Theorem of taste” that allows you to calculate the taste-distance between wine and beer if you know the distances between beer and water and water and wine. Taste impressions don’t have geometrical structure or any comparable kind of structure. And if we didn’t have self-motion then sight would be like that as well.

There’s a passage in Rousseau’s Emile that fits this perspective. It goes like this:

“It is only by our own movements that we gain the idea of space. The child has not this idea, so he stretches out his hand to seize the object within his reach or that which is a hundred paces from him. You take this as a sign of tyranny, an attempt to bid the thing draw near, or to bid you bring it. Nothing of the kind, it is merely that he has no conception of space beyond his reach.”

So imperfect capacity for self-motion goes with imperfect understanding of space, it seems, in the child. Of course Rousseau was writing long before Poincaré. I used Poincaré as the point person for this perspective about the role of self-motion in geometry but indeed the basic ideas go back centuries before. Poincaré explains his view very well in his book La Valeur de la Science of 1905. But that’s the culmination of a tradition of more than two centuries.

For example, many philosophers had debated the following question: Suppose a person who has been blind all their life has an operation that makes them able to see. Can they then, from visual impressions alone, tell for example a cube from a sphere? They already knew the difference by touch, but could they then automatically make the connection between that and sight, or would they have to learn to recognize things by sight through experience?

This is the so-called “Molyneux’s question.” Molyneux raised it in 1688. Obviously it has a lot to do with the question of whether geometry is innate, or whether it is learned by experience.

This thing about a blind person becoming sighted was not just a thought experiment. It could be done through surgery in some cases. Let me read to you a report of the experiences of such a person. This is from the Philosophical Transactions of 1728. A boy who was 13 years old and had been blind all his life got his sight back through a surgical procedure. And his reactions were as follows.

“When he first saw, he was so far from making any Judgment about Distances, that he thought all Objects that he saw touch’d his Eyes, (as he express’d it) as what he felt, did his Skin.”

“He knew not the Shape of any Thing, nor any one Thing from another, however different in Shape, or Magnitude; but upon being told what Things were, whose Form he before knew from feeling, he would carefully observe them, that he might know them again; but having too many Objects to learn at once, he forgot many of them. One Particular only (tho’ it may appear trifling) I will relate; Having often forgot which was the Cat, and which the Dog, he was asham’d to ask; but catching the Cat (which he knew by feeling) he was observ’d to look at her stedfastly, and then setting her down, said, So Puss! I shall know you another Time.”

“He was very much surpriz’d, that those Things which he had lik’d best, did not appear most agreeable to his Eyes, expecting those Persons would appear most beautiful that he lov’d most, and such Things to be most agreeable to his Sight that were so to his Taste.”

“We thought he soon knew what Pictures represented, which were shew’d to him, but we found afterwards we were mistaken; for about two Months after he [became sighted], he discovered [that] they represented solid Bodies; when to that Time he consider’d them only as Party-colour’d Planes, or Surfaces diversified with Variety of Paint; but even then he was no less surpriz’d, expecting the Pictures would feel like the Things they represented, and was amaz’d when he found those Parts, which by their Light and Shadow appear’d now round and uneven, felt only flat like the rest; and ask’d which was the lying Sense: Feeling or Seeing? Being shewn his Father’s Picture in a Locket at his Mother’s Watch, and told what it was, he acknowledged a Likeness, but was vastly surpriz’d; asking, how it could be, that a large Face could be express’d in so little Room, saying, It should have seem’d as impossible to him, as to put a Bushel of any thing into a Pint.” (That is to say, a larger volume into a smaller.)

That’s quite entertaining but also quite significant evidence for the debates we have been considering. Clearly, learning the geometry of sight was a bit like learning a language for this person who became sighted. He didn’t immediately understand the geometrical structure of visual impressions, so clearly all of that is not completely innate. So it speaks against a Kantian account that takes Euclidean geometry to be a precondition of any geometrical thought or geometrical sensory perception.

But the story of the boy who became sighted fits quite well with a Poincaré-type account in which the geometry of sight can only be developed gradually through experience and coordination with self-motion.

Nevertheless, you can still say that Kant was right in a way. Poincaré is in a sense neo-Kantian. According to Poincaré, Euclidean geometry is not innate, but some geometrical notions are. The mind is predisposed to discern geometrical aspects of its surroundings. Hardwired into the mind are not all of Euclid’s axioms but still a good bit of geometry, such as the categorisation of which perceptions are related to geometry at all, and perhaps related to this some concepts such as displacement, rotation, and so on.

So, those are the ways in which geometry is like language. Both are part innate and part shaped by the environment. To adopt a particular language or a particular geometry is to fit your thoughts into an arbitrary and subjective framework. But that’s a good thing because there are no objective frameworks, and without some such conceptual framework, thinking could never even get off the ground in the first place.