This chapter acts as an introduction to every concept that mathematicians intrinsically know but never say out loud that I could think of. The ultimate goal of this chapter is that you should come out the other side with enough awareness of what mathematicians are doing that you could read a math book, namely, my math book.
(11 PDF Pages) A philosophical discussion of what a theory does for us, what mathematical thought is as a category for theories, and how to read the language of mathematical texts.
(28 PDF Pages) An introduction to dependent type theory and a basic discussion of proofs-as-programs, illustrating the reasoning style of propositional logic in a programmatic way.
(24 PDF Pages) in draft 1.5 stage. a mostly philosophical engagement on mathematical thinking + some basic set theory and what we mean by equivalence in a non-constructive world. Towards the end we tie off some notational loose ends and explicitly specify some more conventions.
In this chapter we discuss some topics from real analysis with a focus on how these topics reflect on the nature of $\mathbb{R}^n$ as a space and its deeper, stranger, properties. Throughout the chapter, we develop tools to study mathematical objects in $\mathbb{R}^n$ as well as increasingly describing properties you'd never thought to point out, and discussing what happens in a space without that property. In this way, we slowly build up what the real numbers are from what they would be if they were not.
(17 PDF Pages) A discussion of the properties of real numbers as derived by rewrites on their axioms, as well as our earliest focus on the concerns and mentality of real analysis.
(19 PDF Pages) Here we introduce the notion of limits on sequences, taking great care to both establish them as formal objects and as intuitive ones. In this section we begin the process of thinking about the study of limits as the study of what happens when you look very close to a point. For our Section appendix we discuss the similarities between real numbers as a number system and convergent sequences, and how limits preserve these similarities.
(15 PDF Pages, no appendix yet) In this section we formally discuss intervals in $\mathbb{R}$, introducing notions of open and closed intervals, and open and closed sets. This will be our very first little taste of (non-algebraic) topological concerns, and accordingly we introduce the topological limit characterization, before paying off earlier promised theorems such as Bolzano-Weierstraß and the convergence of Cauchy sequences. For our section appendix we give a much more informal discussion on cardinality, explaining uncountable infinity and when we need to be concerned about it.
(WIP)
Since this text is serialized, it is of course likely at any point that topics that I intend to cover which may be of interest to readers are not yet written about. In the interests of forecasting whether this text will be for you in the near future, the following is a loose plan of unwritten chapters and sections which are planned.
Anatomy of $\mathbb{R}^n$: A Brief Introduction to Real Analysis
Following our discussion of limits, we proceed to construct basic versions of calculus operations, demonstrating both how effective our theory of limits is at simplifying proofs, and some early inklings that the derivative and integral are much more sophisticated than we may realise and made simple only by the fact we use them in $\reals$.
Wrapping our discussion of limits in $\reals$ with a bow, we cover series limits and limits of sequences of functions.
We begin the abstracting of already described concepts such as open and closed sets and intervals to metric spaces, giving their multidimensional versions. It will become clear that most of our constructions around limits abstract immediately to higher dimensions.
This will also be an opportune time to discuss point-set topology, since our metric spaces also define a choice about our notions of closeness, and how that defines the limit. In preparation for what is to come, we also introduce the idea of a topological manifold as a corresponding object to the quotient topology in constructions of non-trivial surfaces.
Our abstraction of a notion of a space continues to elementary measure theory. We develop the notion of volume measuring functions, or measures, and measure spaces. Here we have the opportunity to clear up some common misconceptions about the Lebesgue integral as well as identify similarities between $\sigma$-fields and the field axioms of $\reals$. In the appendix, we explain a little about how measure theory formalizes the theory of probability.
Chapter 3 will focus on Algebra, culminating in a focus of "Anatomy of Abstract Spaces". As I have only outlined a loose plan for these bullet points, they are likely to take up more than one section each, or be reordered.
We begin with a basic discussion of group theory, induced by monoids on alphabets and constructed with generator and relation sets. Our principle goal in this section is to introduce the notion of groups, group homomorphisms, subgroups, normal subgroups, central subgroups, abelian groups, and quotient groups, touching briefly on group actions. An appendix discussing formal languages may also appear here.
We use our notion of formal groups to begin a discussion of vector spaces. By using the structure of groups as the backbone, we begin a discussion of matrices as explicitly linear operators, the group homomorphisms of vector-spaces-as-groups. This will assist us in constructing notions such as the null space, and the quotient space. In this way we cement linear algebra as a principly algebraic discipline. Our discussion will of course not be complete without a study of eigenvalues and eigenvectors.
We begin with a very light discussion of category theory, which we now have some background to discuss as we have seen the categories of sets, groups, and vector spaces. Our principle goal here is to extract the notion of dual spaces, which we use to begin a study of different kinds of tensor algebras, in particular the exterior algebra, and better motivate the matrix determinant.
We spend a bit of time talking about representations with the interest in separating algebraic constructions from their parameterised counterparts which we may be familiar with as column vectors or matrix linear operators. That is, we are interested in giving character to the idea of a thing which is distinct and has function yet is independent of how we describe it. This will probably also be a good place to continue our discussion of group actions, and consolidate some discussion of matrix groups. This exposition may need to be intermingled or reordered with our discussion of tensors.
Chapter 4 will focus on concepts that underlie undergraduate applied mathematics, paving the way for our discussion of differential geometry in chapter 5. We continue our discussion of (multivariable) calculus and real analysis, in particular Hilbert spaces, culminating in a proof of Picard-Lindelof, which we springboard off of into a discussion of differential equations. We will also have to take this opportunity to begin a discussion of classical mechanics (i.e. prequantum physics) and perhaps symplectic structure, which will help motivate much of what we do later on in differential geometry and control theory. It is possible in this chapter that we will introduce finite difference methods and finite element methods, depending on whether it seems thematically appropriate at that point, and or discuss stochastic differential equations.
Chapter 5 begins our discussion of differential geometry in earnest. By the point I am writing this I hope to have developed a narration of how the concerns of differential geometry as well as the structure provided by it remain supremely relevant even to the engineer who is only concerned with $\reals^3$. This chapter will draw on all previous chapters, synthesizing an understanding of (co)tangent functors, flows, connections, Lie groups, and may culminate in a proof of symplectic reduction theorem. We may also be forced to briefly return to topology for a description of cohomology, from which the nature of the generalized stoke's theorem will become obvious to us (putting the chapter at risk of being split into two). This chapter is also likely to be written around the conceit that differential geomtry is as much a study of differential equations as it is of geometry, identifying the two discussions with one another and advocating for the use of differential forms in multivariable calculus.
Chapter 6 will be where dragons I am yet to fully conquer lurk. I cannot say for certain what it will be about, but possible options are control theory, rewrite systems, numerical methods, or (dare I jinx myself) fluid dynamics. In a far less ambitious direction I could discuss quantum mechanics, but the danger then is that I have to go learn about quantum field theory properly this time.