﻿I’m going to explain everything from first-principles, but very lightly.


You start with propositions. The study of what is true/false. After that you move on to defining the rules of the rules of the game: axiomatic systems.


Having done that, you talk about a specific rule set that 99% of mathematicians use called ZFC. It basically defines set theory, while avoiding paradoxes. One thing we have a hard time getting rid of is the axiom of choice paradox, where on one hand it’s needed to prove that every vector space has a basis, but it also has unfortunate consequences like the banach tarski paradox, where basically a sphere can be duplicated without any problem coming up.


Now that we have sets, we do a little bit of meta study: the classification of sets. This meta study hints at the idea of the mega mega study: category theory.


Next, you define the numbers as we know them. You start with the natural numbers N, and manually defined addition and multiplication using add1 and sub1.


Next you use quotient sets to define the integers, and also define plus and times. You make sure these are well defined.


Similarly, you defined the rationals, and the reals, along with their operators.


Next you start talking about topological spaces, because physics needs continuity, and topological spaces are the minimal structure needed to talk about continuity. You defined what a continuous map is in terms of open sets on the domain and target.


You also revisit meta stuff by looking at the (not) classification of topological spaces from it’s invariants. Again you also get a hint at the meta meta stuff of category theory, but not explicitly.


Next you introduce manifolds, which are essentially topological spaces with the extra condition that they’re locally euclidean. We also immediately talk about bundles, since those will end up being very relevant to define vector fields later on. Bundles are very simple, and it’s essentially where you attach a fiber at every point. Bundles are not actually used for construction purposes, but rather for analyzing the bundle structure itself (for example later on we want to talk about smooth vector fields, which will be discussed in terms of the tangent bundle’s section functions being smooth).


Next we talk about smooth manifolds. A smooth manifold has extra structure: it’s equipped with an atlas, where all the charts are each smooth compatible with each other, which means that any overlapping chart agrees on the differentiability status of every curve on their overlap: or in other words that the chart transition map is smooth.


Our next step is to introduce vector spaces and tensors, because they obviously become very important later, since many things have a vector space structure, and tensors can be used to encode a lot of information into a tiny object, and make equations very compact.


Next we talk about tangent spaces, and the idea of a “tangent vector”. This idea is important because we want to talk about vector fields, and vector fields are essentially the section of a tangent bundle.


The reason we care about vector fields is that a lot of the partial differential equations that govern shit are written in terms of vector fields satisfying certain properties. This is exactly what happens with the Maxwell equations for example, and also Einstein field equations. I mean field is written in the name right there.