1. Philosophy of Maths 1: Geometry
The Unreasonable Effectiveness of Weird Mathematics

2. Overview
We turn today to look at how geometry interacts with modern physics.
We begin with the idea of reference frames and the transformations between them.
This applies to classical (broadly, Newtonian) physics and also to relativity.
We then look at the idea of configuration spaces and phase space, the fundamental objects of mathematical physics.
This leads us to Noether’s Theorem, which makes a surprising new claim about the nature of physics as a science.

3. Philosophically, try to keep in mind the following questions:
Is the “space” we’re talking about something objectively real or imagined?
Is it arrived at by empirical means?
Does this distinction matter?
Is geometry being used as a tool here rather than a form of knowledge in its own right?
If the former, how can we tell whether it’s being used correctly?
If the latter, what is it knowledge of?

4. Reference Frames

5. Coordinates
Physics is especially concerned with motion, which means it needs a good mathematical handle on points (i.e. locations or positions) in space.
The idea is to pick a point in space that will serve as an origin for all your measurements.
For practical purposes we don’t need to pick a “point”, just something small enough that we can’t measure it as having any size or duration.
But to make the maths simpler we usually work with sizeless points and instants .
Then you describe a location in 3D space by saying how far you need to travel in each of three directions to reach that location from the origin.

6. Reference Frames
Ordinary space is three-dimensional, which means we need three coordinates to identify a location.
Choose an origin, say Centrepoint in London.
Now any point anywhere in the universe can be identified by saying:
Start at the origin.
Move a certain distance East
Move a certain distance North
Move a certain distance directly upwards.
…as long as “east”, “north” and “upwards” are well-defined.
Note that, if you were an observer on Mars, the origin of this coordinate system would appear to be moving – it’s rotating with the Earth and also orbiting the Sun.
But to an observer on the Earth, it appears that Mars is moving while the origin at Centrepoint stays still.

7. Reference Frames A choice of coordinates is called a reference frame.
Two scientists might easily study the same phenomenon but within different reference frames.
Their frames might even be moving relative to each other. For example, one scientist might be on a moving train while the other stands on a platform.
And note that both of these scientists are on the Earth, which is both rotating and orbiting the sun at enormous speed, and the sun itself is orbiting the centre of the Milky Way galaxy.
There is no “privileged” reference frame.

8. Inertial Frames A reference frame might be accelerating.
This does not mean it’s moving; it just means a force is acting on the frame. For example, a reference frame based at a point on the surface of the Earth is experiencing acceleration due to gravity. It is also being accelerated by the rotation of the Earth and its orbit around the sun.
Non-accelerating reference frames are called inertial.
Any inertial frame viewed from another inertial frame is either stationary or moving with constant velocity – i.e. not accelerating.

9. Reference Frames
A problem: if two scientists study the same phenomenon from within different reference frames, they will usually get different results.
The laws of nature shouldn’t depend on how you’re looking at them! At least not if we’re trying to describe the universe rather than just our subjective experience of it.
We say physics should be coordinate independent.
Does this mean physics is hopeless?
Not if there is a mathematically tractable way to transform your observation (in your reference frame) into the “same” observation in my reference frame.

10. Galilean Transformations
Classical physics assumes the geometry of physical space is Euclidean.
There are a family of “translations” between reference frames in this space called Galilean Transformations.
Intuitively, they are rotations and rigid movements.
In the language of linear algebra, these are “affine transformations”, which makes them very well-behaved.
Classical physics can study whatever is unchanged by Galilean transformations.

11. Lorentz Transformations
In relativity, we have to change the formulae we use to translate between reference frames.
Fundamentally this is because of the need for the speed of light to be a maximum speed that measures the same in every reference frame.
The result is the Lorentz transformations.
But if we assume space has Euclidean geometry, the result isn’t good.
In particular, it means “being an inertial reference frame” depends on your reference frame, which in turn causes the whole project of physics to collapse: we can no longer agree on who should agree on physical facts!
But if we change to hyperbolic geometry, everything works out nicely.

12. The Lorentz transformations are based on a way of measuring distance called the Lorentz Metric.
The shortest Lorentz distance between two points is the line where a plane going through the origin cuts the hyperboloid.
It turns out that this measure of distance is the right one to use to make relativity work properly. But that produces a hyperbolic geometry, which seems to mean that the space we live in is a hyperbolic space…

13. Configuration Space

14. Our points needn’t be in “ordinary” physical space.
For example, each of us occupies a point in this height-and-weight space (ignore the colours).
Each of the coordinates is a number, and one can’t be figured out from the other so we need both to find our location. That makes it a 2D space.
But the points are only arranged in a physical space because we made a picture out of them. But the picture is quite useful, and its geometry can yield interesting facts.

17. Here are two simulated movements of a “double pendulum”.
Their starting conditions differ only very slightly, but they rapidly develop very different behaviour.
It seems chaotic, even though the system is quite simple.

18. Viewing the same kind of motion in phase space reveals an underlying symmetry and even a kind of simplicity.

19. To describe the way an aircraft’s moving at particular moment we need six numbers:
X says how fast it’s going in the forwards direction
Y says how fast it’s moving sideways (e.g. due to crosswind)
Z says how fast it’s moving vertically
R says how fast it’s rolling
P says how fast it’s pitching
Y says how fast it’s yawing
These six coordinates are all independent of each other and each can easily be represented by a number. So the “configuration space” of a flying aircraft is 6-dimensional.
An aircraft at each moment occupies a point in its configuration space. If it is travelling at a constant speed then it stays at the same point in configuration space.

20. Noether’s Theorem

21. Symmetries in Phase Space
Noether’s Theorem says roughly this:
Every symmetry of the phase space portrait of a system corresponds to a law of physics for that system.
(This is very rough)
Specifically, there is a correspondence between conservation laws and geometric symmetries.
This suggests that when we do physics, we’re really doing geometry, albeit not usually that of physical space but one generated by the physical phenomenon itself.

22. Example: Quantum Mechanics
In the 1950s, John von Neumann provided a mathematical formulation of quantum mechanics that remains the most popular today.
It requires some complicated machinery but again involves constructing a space whose points represent the states a system can be in
This space is very abstract; it is infinite-dimensional and “projective”, which means it doesn’t look anything like Euclidean space.
The symmetries (in a special sense) again represent physically meaningful “facts”.

23. Philosophical Questions
What should count as “a space”?
Are we free to impose any geometry we choose on a space (as in the choice of the Lorentz metric)?
Is geometry validated by its practical usefulness alone? If so, is it still knowledge?
Can physics be reduced to a search for geometric symmetries?
What about theoretical physics, which in some cases arises solely from such considerations and produces hypotheses that might never be empirically testable?