Subject: Mathematics
Thoughts on Mathematical Induction: Part I
Induction, even setting aside its raison d'être as an inferential tool for mathematical proofs, is a remarkably elegant concept. Most people first encounter it indirectly through recursion in programming language lessons, and in my personal experience, rarely explore any further. Pinning down the scope for this series initially perplexed me. It's best we start in known territory and gently extend outward. To that end, I will only assume that the reader is capable of basic procedural and algorithmic reasoning and that they're either curious or motivated. We'll approach induction by first understanding recursion, since that's how many encounter it. Some confusion is normal (and inevitable); repeated encounters with these ideas will clear most of it in time.
Developing Indus
Nearly a decade ago, I stumbled upon Peter Shirley's Ray Tracing in One Weekend series and it completely captivated me. The only problem was that I had no exposure to computer science, programming, or practically any related domains. At the time, I was an audio engineer and a project like this felt a distant dream. I'd revisit the book occasionally, marvel at all the cool things and that was that.
In 2018, I took my first real step by enrolling in a game-development undergraduate program. It was a mixed bag by any metric. The programme was not standardized nor was it rigorous (and I’m being charitable). What it did do was introduce me to some amazing people who were a decade younger but accomplished fantastic things. It was brutally humbling and still is. I learned a lot of ad-hoc C++, some Unity, and Unreal Engine was my engine of choice and where I focused on the most. Scrambling in the trenches past the barrage of tests and other tribulations on the personal front left very little time for any real stock-taking or extracurricular learning.
Modern Mathematical Logic
Introduction The Nature of Mathematical Logic Mathematical logic originated as an attempt ░░░░ codify ░░░░ formalize:
The language of mathematics. ░░░░ The basic assumptions of mathematics. The permissible rules of proof. ░░░░ of ░░░░ successful ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ study mathematical ░░░░ and ░░░░ ░░░░ mathematics itself. For ░░░░ ░░░░ will eventually give ░░░░ ░░░░ mathematical ░░░░ ░░░░ a ░░░░ ░░░░ ░░░░ to ░░░░ confusion with our ░░░░ ░░░░ understanding of ░░░░ ░░░░ proof ░░░░ we will ░░░░ ░░░░ objects deductions. ░░░░ can ░░░░ ░░░░ ░░░░ eventual ░░░░ of ░░░░ deduction as ░░░░ ░░░░ ░░░░ precise ░░░░ definition of continuity, ░░░░ replaces the ░░░░ “a graph that ░░░░ be drawn without lifting ░░░░ pencil.” ░░░░ we have ░░░░ ░░░░ notion ░░░░ ░░░░ ░░░░ we will ░░░░ ░░░░ ░░░░ into ░░░░ mathematical objects, ░░░░ us ░░░░ prove mathematical ░░░░ about ░░░░ ░░░░ ░░░░ mathematical reasoning. For example, ░░░░ ░░░░ open up ░░░░ ░░░░ ░░░░ ░░░░ that ░░░░ ░░░░ no deduction of ░░░░ mathematical ░░░░
The Tao of Types
Simple Types Before ░░░░ dependent types with ░░░░ ░░░░ ░░░░ ░░░░ we’ll take ░░░░ ░░░░ ░░░░ something ░░░░ ░░░░ ░░░░ simple types░░░░ starting ░░░░ ░░░░ ░░░░ ░░░░ type of all, the functon type $\_\to\_$. ░░░░ ░░░░ ░░░░ ░░░░ types is that ░░░░ ░░░░ not ░░░░ what to ░░░░ ░░░░ We’ll ░░░░ this with ░░░░ which work ░░░░ any ░░░░ called $\textsf{Set}$ in Agda. That is, ░░░░ will ░░░░ polymorphic ░░░░ without introducing a ░░░░ ░░░░ ░░░░ ░░░░ which ░░░░ ░░░░ do later once we have dependent ░░░░ in our arsenal. ░░░░ of polymorphic constructions ░░░░ ░░░░ $\textsf{id}$ and composition $\_ \circ \_$ ░░░░ ░░░░ ░░░░ a first ░░░░ of ░░░░ ░░░░