Einstein’s theory of special relativity defines time and space operationally, that is to say, in terms of the actions performed to measure them. This is analogous to the constructivist spirit of classical geometry.

Transcript

Oh no, we are chained to a wall! Aaah! This is going to mess up our geometry big time. Remember what Poincaré said: self-motion is the essence of geometry. We understand that part of the environment to be geometrical that we can cancel through self-motion, through a change of perspective.

Suppose you are looking at a chair, let’s say, and somebody tips it over so that it’s laying on its side, or somebody moves it to the other end of the room. Those are geometrical transformations: rotations and displacements in space. They are the equivalence relations of space; the isometries: things you can do without changing metric relationships.

You know that these are geometrical equivalence transformations because you can cancel them through self-motion. When the guy knocks the chair over, you can tilt your head 90 degrees, and you have restored the original visual impression of the chair. And if the guy moves the chair five meters that way, then you yourself can move five meters in the same direction and once again the chair makes precisely the same visual impression on your retina as it did before.

This is how you know that rotations and displacements are geometrical equivalence transformations. The more you accumulate experience with these kinds of scenarios, the more you begin to grasp the group of geometrical transformations as a whole. You get a global sense of what kinds of transformations are possible, how they combine and interact, and so on. This process might lead you to Euclidean or non-Euclidean conceptions of space depending on your experiences. You get to know space and what kind of geometry it has by getting to know its transformation group: that is to say, what kinds of rotations and displacements exist, what happens if you do one after the other, and so on.

Now, what about the scenario when we are chained up? We must imagine that we have been chained to this wall for life. We don’t know any other reality than this.

Our sense of what geometrical transformations are possible will be very different. There is still geometry because there are still visual impressions that we can cancel through self-motion. If an object is moving across our field of view, we can keep the retinal impressions the same by tracking it with a motion of our eyes. So we understand the geometry of sideways motion well since we can move our eyes from left to right, or point our gaze in different directions.

We also understand the geometry of depth to some extent. If an object is moving away from us, we can keep track of that through self-motion also, but of a very different kind. They eye has a lens in it. The curvature of the lens is variable and is controlled by a muscle. Depending on whether you need to focus on objects that are near or far, the muscle will pinch or pull the lens so that it is more round or more flat in order to have the right focal distance for the object you are looking at. In this way you can keep track of how much an object has moved in depth by recording how much the lens needs to be adjusted to restore focus. So this gives you the data to develop a geometry of depth.

So our chains do not deprive us of geometry altogether. We can still develop the geometry of width and the geometry of depth. But these are separate geometries to us. A free person will know that width and depth are merely two dimensions of the same kind of thing. They are both spatial dimensions. They are interchangeable and homogenous. The free person will know that since they can turn width into depth by self-motion. They just need to go stand over there and the old width is the new depth and vice versa.

But we who are chained are deprived of this experience. So to us width and depth remain qualitatively different kind of things altogether. Indeed, we measure distance in width and distance in depth completely different units. We count distance in width by the direction in which our eyes are pointing, so the unit is degrees for example. An object is 30 degrees to the left of another, for example, we might say. But we count depth by how much the lens needs to be bent to achieve focus. So the unit is something like a unit of force corresponding to the muscular effort involved. That’s a completely different kind of thing altogether, and cannot be compared with our degree measures that we used to quantify position in the width direction.

It’s not so strange that width and depth would be qualitatively different things. You already treat various measurements of the same object as qualitatively different in your everyday life. For example, suppose somebody asked you: Is this building wider than it is old? Of course that doesn’t make any sense. You cannot compare a distance in space with a duration in time. Because those quantities are determined in fundamentally different kind of ways, they are measured in completely different kinds of units, and so on. Well, just as you think time and space are not comparable, so the chained person thinks depth and width are not comparable. Samesies.

In fact, maybe you are are just as delusional as the chained guy, and for much the same reason. Actually time and space are a lot more comparable and interchangeable than you think, as Einstein’s theory of relativity says. We don’t realise this in our everyday experience, because relativistic effects become significant only at high speeds, somewhat close to the speed of light. Compared to the speed of light you have practically been standing still your whole life, even when flooring it on the highway. So you might as well have been chained to a wall. The sum total of all your visual and sensory impressions are severely and systematically impoverished just like the guy chained to a wall. Just as he doesn’t realise the fundamental unity of width and depth, so you don’t realise the fundamental unity of time and space. And for the same reason: you are both essentially standing still.

I took this example from Feynman’s famous lectures on physics. Why don’t we listen to his version as well? The classic Feynman lectures on physics are nowadays available for free at a Caltech website, audio recordings and all.

“When we look at an object, there is an obvious thing we might call the ‘apparent width’, and another we might call the ‘depth’. But the two ideas, width and depth, are not fundamental properties of the object, because if we step aside and look at the same thing from a different angle, we get a different width and a different depth, and we may develop some formulas for computing the new ones from the old ones and the angles involved. … If it were impossible ever to move, and we always saw a given object from the same position, then this whole business would be irrelevant—[width and depth] would appear to have quite different qualities, because one appears as a subtended optical angle and the other involves some focusing of the eyes …; they would seem to be very different things and would never get mixed up. It is because we can walk around that we realize that depth and width are, somehow or other, just two different aspects of the same thing.

[In Einstein’s theory of special relativity] also we have a mixture---of positions and the time. … In the space measurements of one man there is mixed in a little bit of the time, as seen by the other. Our analogy permits us to generate this idea: The ‘reality’ of an object that we are looking at is somehow greater (speaking crudely and intuitively) than its ‘width’ and its ‘depth’ because they depend upon how we look at it; when we move to a new position, our brain immediately recalculates the width and the depth. But our brain does not immediately recalculate coordinates and time when we move at high speed, because we have had no effective experience of going nearly as fast as light to appreciate the fact that time and space are also of the same nature. It is as though we were always stuck in the position of having to look at just the width of something, not being able to move our heads appreciably one way or the other.” (I.17-1)

I love this thought experiment with the chained guy. Plato’s cave 2.0. And it is perfect for our purposes today. This is going to be the concluding episode of my history and philosophy of geometry story arc, and the theme will be how everything goes full circle and the beautiful ideas from days of old are as relevant as ever to us self-absorbed moderns as well. The guy chained to a wall is a perfect backward-looking example, and a perfect forward-looking example. Back to the operationalism of Greek geometry, and forward to Einsteinian modernity.

We started, way back when, with the Greeks and their ubiquitous ruler and compass. Always with the making, those guys. Lines and circles are nothing but the things you get when you draw with these tools. Not abstract things, not axiomatically defined things. Lines and circles are operations. They are things you do.

The Greeks realised that this was the rigorous way to do mathematics. The epistemological humility of the maker is far superior to hubris of the philosopher who think they can concoct a perfect theoretical system in the abstract using the power of their mind alone. People are not as good at that as they think. Time and time again, somebody’s pretentious abstract theory has proved to contain various unintended contradictions and unnoticed assumptions. As the Greeks knew all too well: the works of Plato and Aristotle do little else than poke holes in other people’s bad theories. So we should stop trying to philosophise about essences