This serialized book, or perhaps blog, or perhaps online textbook, this text, is my attempt to thoroughly review what I know about mathematics with a fine toothed comb and present it much more intelligibly than I received it. In the process of this, I hope to examine gaps in my knowledge, expand it to its logical conclusions, and make what I learn comprehensible to as large an audience as possible.
The following reflect my motivations for writing this text as well as some notable qualities it may have due to those motivations.
This text will describe the concepts I am concerned with to my friends, many of whom are oriented to at least one of software or literature. As such, this text will endeavor to hold the reader's hand at times and ensure that ideas are legible to someone with only shallow mathematical background.
In the interests of the above, this text will be opinionated about how the reader should think about many concepts. In mathematics texts often present ideas in formalism and make it the duty of the reader to develop a mental framework, but where possible, I will assert a choice of mental framework and work to make these choices cohesive with one another. Consequently there are many ways in which this text will be non-standard in its presentation, and I make no apology for this.
In order to keep a balance between a comprehensible text and an approachable one, subsections of chapters may be marked as section appendices, which may hold important proofs but the reader is encouraged to return to later if they do not understand it or its motivations at first. Additionally, proofs and examples are written relatively verbosely and made minimizable when their fine details are not of importance to the narrative flow.
I have chosen to proceed with two particular alterations to the standard mathematical pedagogy which will appear throughout the text. First, that we focus on a functional description of mathematics, not treating it as an absolute but rather as a framework which allows us to structure our operations, including things like applying theorems or constructing objects. Second, I will often use the symbol $\square$ in this text to denote an anonymous function, e.g. $f(\square) = f$, rather than the usual $f(\space \cdot \space)$ since this notation will become greatly valuable for aggressively emphasizing the construction of functions.
Finally, this text is a living document in that it will be edited as necessary and upon reasonable request, criticism, or suggestion. Indeed it is in a sense a publically available draft that will allow me to write this text without keeping a partially written version to myself when it may prove useful incomplete. And no matter how much I know about a topic, it is not uncommon that I learn something new which changes my perspective about how best to exposit it, or indeed realise that it is necessary to explain things in a different order.
What is Application Unification? I am by training a pure mathematician, but I have become surrounded by a world of applications that constantly surprises me with the richness of its structure and the jankness of its loose use of formalisms. In my mind, this is the perfect opportunity to simply invent new, better formalisms, a project which I will embark upon cautiously. I am becoming an applied mathematician, and you're coming with me.
Finally, a disclaimer. I am in no way an expert, in the sense of the standard that I have come to understand truly defines an expert in mathematics, on any of the topics I write about here. My expertise is relative, and I am writing this text in part to expand that expertise that I do have. In the words Russel O'Connor in his copyright notice in the Coq-Corn package,
"THE PROOF IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED"