Non-Euclidean geometry does not prove Euclid was wrong

Possibly the most misapplied data in philosophical discussions is the fact of non-Euclidean geometry. The usual story is that for two thousand years, it was believed that Euclid’s Fifth Postulate was true and self-evident, but then the development of non-Euclidean geometry (or possibly the use of non-Euclidean geometry in a physical theory) proved that Euclid’s Fifth Postulate is not true.

This story has been used to counter the notion of self evidence as a source of knowledge and it has been used against Kant’s notion of space as an intuition. However, the claim is false. The discovery of non-Euclidean geometry had no bearing on on the truth of Euclidean geometry (although it did prove that the Fifth Postulate is independent), and because non-Euclidean geometry has no bearing on the truth of Euclidean geometry, its use in a physical theory, even a true physical theory, also has no bearing on the truth of Euclidean geometry.

Let’s start with a theory of containers. I want to develop a theory that lets me ensure that everything I need has been packed for my vacation. I start with a collection of containers, many of which can go in other containers. The largest container is my car, inside that is a set of suitcases, inside the suitcases are several smaller bags like a shaving kit, and inside those are other things like a case for my razor. All of that can be quite daunting, so I want to come up with a mathematical theory to make sure I have everything packed.

Let c1, c2, etc. represent a container of any sort, and let < represent the relation of being inside a container. Then I propose a theory of containers that says that if container c1 is inside container c2, and container c2 is inside container c3, then c1 is inside c3:

Gudemanian theory of containers

c1<c2 and c2<c3 implies c1<c3

Using this theory, I can be sure that if my razor is inside its case and the case is inside my shaving kit, and the shaving kit is inside my suitcase, and my suitcases is inside my car, then my razor is inside my car. Makes sense, right?

Not so fast! A house is a container, isn’t it? And some houses contain a central outdoor courtyard which is also a container, but things in the courtyard are not in the house. In this case, if c1 is a suitcase in the courtyard and c2 is the courtyard and c3 is the house, then c1<c2 and c2<c3, but it is not the case that c1<c3. We have a non-Gudemanian theory of containers! Not only that, but the non-Gudemanian theory of containers can be a part of a true physical theory as my example demonstrates.

Does this prove that the Gudemanian theory is wrong? Of course not. All it proves is that there is a different theory of containers that deals with different sorts of things that might be called containers. We have two different theories dealing with two different types of objects, so the truth of one has no bearing on the truth of the other.

The things that the theory works on is called its model. Note that a single theory can have many models, some of which are completely unrelated to the intended model. For example, the Gudemanian theory of containers also has numbers as a model. Let c1, c2, etc. represent numbers and let < represent the less-than relation. Then with this model, the Gudemanian theory is still true. Another possible model might be preference: I prefer chocolate ice cream to vanilla ice cream, and I prefer vanilla ice cream to spinach, so does this mean that I prefer chocolate ice cream to spinach? It probably does! But maybe not. The question of whether preferences follow the Gudemanian theory of containers (another way to say this is that preferences are transitive) is an interesting question of human action, but let’s not go there for now.

Non-Euclidean geometry works exactly the same way as non-Gudemanian container theory. To be more precise, here are Euclid’s five axioms:

  1. Between any two points you can draw a line.
  2. Given any line, you can extend the line in any direction as far as you want.
  3. Given any line, you can draw a circle with one end as the center and with the line as the radius.
  4. All right angles are equal.
  5. If two lines x and y intersect a third line z in such a way that on one side of z, the angles sum to less than two right angles, then x and y will intersect on that side if they are extended far enough.

You can see why so many mathematicians have always been suspicious of the Fifth Postulate: it is a lot more complex than the others, and seems less intuitive. Various mathematicians tried to prove the Fifth Postulate from the other four, but no one was successful.

Eventually someone proved that there is a model of a theory that uses Euclid’s first four axioms plus a fifth that is incompatible with Euclid’s Fifth. To get a model, they restricted the plane to the interior of a circle and replaced lines with chords of the circle, and changed what an angle is. This system followed all of Euclid’s axioms except the Fifth. Clearly, however, this is did not prove that Euclid’s Fifth is wrong; it proved that Euclid’s Fifth is independent–that is, it proved that Euclid’s Fifth could not be proved from the other four.

A century ago, when this work was going on, it was not fully appreciated that a theory can be viewed as an abstract thing independent of its model, and that a single theory can have many models. In fact, Euclidean (plane) geometry has, in addition to the intended model, a model where a “point” is a pair of real numbers, and a line is a set of points satisfying what is called linear equation (for the obvious reason). This model is called Cartesian geometry or analytic geometry, and it is what you studied in your calculus classes if you had them. Back when analytic geometry was developed, it was not really appreciated that they were developing an alternate model of Euclidean geometry; they thought they were just using arithmetic to describe geometric entities.

But today we know that most theories with one model have many models. We use the existence of a model to prove that a theory is consistent. We change an axiom and show that the resulting theory has a model to prove that the axiom is independent. In logic you generally want all of your axioms to be independent of each other. If they aren’t, you may as well remove one that is not independent and turn it into a theorem instead.

Our understanding of theories and models today completely undermines all of the metaphysical implications that were once thought to attach to the existence of non-Euclidean geometries. It turns out that these non-Euclidean geometries are not remarkable at all; they are just other theories with other models. No one has ever demonstrated a set of straight lines in a plane that violate Euclid’s Fifth Postulate; all they did was prove that the Fifth is independent.

What about the use of non-Euclidean geometry in General Relativity? Doesn’t that prove that space is really non-Euclidean? This is a difficult claim to assess because I’m not sure what it would mean for space to “really” be non-Euclidean. I can only imagine that we are talking about some theory of space as a real, independently-existing entity that follows the laws of Euclid like matter was once thought to follow the laws of Newton. Let’s call this Euclidean realism, and the contrary let’s call anti-Euclidean realism. I don’ t know of any specific metaphysical claims of Euclidean realism, but there was lots of talk that seemed to assume it, so let’s assume it was a real position.

Assuming that General Relativity is true, then since General Relativity is formulated in terms of non-Euclidean geometry, this proves that space is truly non-Euclidean, right? Well, not really. The problem is that General Relativity can also be formulated in Euclidean geometry. You see, the non-Euclidean geometry used in General Relativity does not describe a space different from Euclidean space, rather it describes the same space using geodesics rather than straight lines. As one would expect, it is just another model.

Now, I don’t mean to discount the value of geodesics. Recall that in Newtonian mechanics, an object always moves in a straight line at a constant velocity unless it is acted on by a force, where one possible force is gravity. A geodesic is defined in such a way that any object always follows a geodesic unless it is acted by a force other than gravity, so an object in free fall is following a geodesic; in other words, gravity is treated, not as a force, but as a component of the shape of space.

Which is very cool, and there are good reasons for it, which I won’t go into, but the important point is that this just a different model from the Euclidean one; it doesn’t show that the Euclidean model is false. So, not only does non-Euclidean geometry not disprove Euclidean geometry, General Relativity does not disprove Euclidean realism.

One more point: none of this works against Kant’s theory of space either. Kant was not a Euclidean realist because he wasn’t any stripe of realist. His position was that space is just the way that our minds interpret the world and that Euclid’s laws are true because of how our minds work. Idealism has acquired a bad reputation from a lot of bad philosophy that followed Kant, much of it implying that since the world is merely an idea, it is ephemeral in the way that dreams are, or that reality is subject to our will. I don’t think Kant would have endorsed these ideas; Kant believed in a real world, just as enduring and just as obstinate in the face of our preferences as the most hard-nosed realist believes in, he just didn’t think that our experiences of the real world are identical with the real world.

One reason Kant believed this is precisely because of the problem of Euclid’s axioms and how we could know them. If Euclid’s axioms are truths about a real, independently existing world, how can knowledge about them seem to be innate? How is that we seem to know these postulates a priori (that is, prior to any experience). It seems rather remarkable, doesn’t it? It’s like knowing innately that warm-blooded, fur-bearing animals nurse their young. How could we know that without investigating?

Kant’s solution is that we can know this because geometry isn’t about the independently-existing world; it’s about our own capacity to experience the world. It seems a lot more plausible that we can inspect our own possibility of perception by introspection than it does that we could know something about the world by introspection. If you can’t imagine what it would look like for two parallel lines to intersect, it seems plausible that you could never perceive two parallel lines intersecting.

In any case, I think I’ve already said all I need to in defense of Kant; since non-Euclidean models don’t prove Euclid wrong, they don’t prove Kant wrong.


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