# The Geometrical Structure of Space and Time

I have been sitting in on Tim Maudlin‘s philosophy of physics course at nyu where we are reading his recent book (and a couple of others). I was planning on going today but in light of the storm that is whopping us I may end up staying in and playing Naruto Shippuden 3 (and possibly watching some of the Naruto series in general). So I thought I might spend a little time reflecting on what has happened so far.

One of the immediate questions that arises is why anyone would ever want or need to posit a geometrical structure for space or time. Using Aristotle and Newton as example Maudlin makes the case that the answer is that they need to appeal to a certain structure in order to make sense of motion. So, for example, Aristotle’s claim that objects made of Earth move down towards the center of the universe naturally leads us to think of the universe as a bounded circle or sphere.

In Newton’s case he needs absolute space and time to have certain geometrical properties in order to make sense of the first and second laws of motion. The first law tells us that an object in motion tends to stay in motion unless it is acted on by an outside force (and an object at rest tends to stay at rest, etc). Similarly the second law tells us what happens when we apply forces to these objects. We will only be able to make sense of acceleration if we have a grip on uniform motion. All of this requires that space have a certain structure. Namely, roughly the structure of Euclidian space. This requires, specifically, that space be immovable and everywhere the same over time. If space is a constant then we can define motion relative to it. Uniform motion is when you cover the same amount of space in equal amounts of time, acceleration a deviation from this.

As Maudlin points out all of this has to do with the geometrical structure of space and time but if one looks at physics as it is done today one does not find it presented in terms of geometry (as Newton himself did) but finds that it is presented in algebraic form. Thus we are used to seeing representations of the second law as F=mA rather than presented in geometrical terms. This is done by associating ordered sets of numbers from the real number line R with points in space and time via a coordinate system. The most easily recognizable coordinate system is Cartesian coordinates. In such a coordinate system we have an axis for each dimension and ordered n-tuples of numbers then can stand for or represent points in space with (0,0,0) as the origin (in a three dimensional space). Maudlin then suggests that though this is useful because it allows us to solve geometrical problems using the tools of arithmetic it obscures the central fact that the arithmetic is merely representing the geometrical properties, and even worse, it can mislead us about those properties by encoding information that is merely arithmetical (see page 27 of his book).

I think this is an interesting and important point. On thing that is kind of suggestive is that this idea, which for the most part as far as I know was initiated by Descartes (I seem to remember a story about Descartes watching a fly buzzing around and getting the idea of graphing functions), might be seen as a kind of precursor to the notion of duality that looks to be playing an important role in contemporary physics. We might say that Newton’s Euclidean and geometric theory is dual to our modern Algebraic/arithmetic theory. It is not the same thing as what we find in string theory where we see that theories with very large distance scales are dual to theories that have very short distance scales nor is it close to the idea that you can capture all of a theory at a higher-dimension in a theory pitched at a lower level.

Another interesting thought, which might be a corollary of the above thought, is that ultimately the arithmetic conception might be right, or at least more fundamental. If space and time are emergent phenomena then the fundamental nature of reality may best be captured arithmetically and the geometrical structure emerge as convenient shorthand at macroscopic distances.

This is all very interesting but it is mostly a digression from the text and the course. At this point we have been talking about the debate between those who think all there is to space are the relations between objects (championed by Leibiniz) those who think that space exists independently of all of these relations (Newton and Clarke). His aim there is to develop, and ultimately try to solve, a puzzle. The puzzle is this. On the one hand we have good reason to think that there are absolute rotations (basically due to Newton’s Bucket), but we also have good reason to think there can’t be absolute motion (it results in postulating motions which cannot be experimentally tested or detected). But then how do we make sense of absolute rotation?

These interesting questions will have to wait so I can check on the weather and take the dog out…