April 17, 2025 |
Section: Notes
Simple Types Before exploring dependent types with all its beautiful complexities we’ll take a look at something simpler, appropriately named simple types, starting with the most fundamental type of all, the functon type $\_\to\_$. One issue with simple types is that it is not clear what to start with. We’ll generalize this with constructions which work for any type, called $\textsf{Set}$ in Agda. That is, we will use polymorphic constructions without introducing a formal account for polymorphism which we will do later once we have dependent types in our arsenal. Examples of polymorphic constructions are identity $\textsf{id}$ and composition $\_ \circ \_$ which give us a first taste of category theory.
March 17, 2025 |
Section: Notes
Part I: Rings The Integers The Well Ordering Principle and Induction Our general goal is the generalization of common algebraic structures; we will try to capture their essence by determining axioms which are responsible for their properties. We start with the set of integers $\mathbb{Z}$, considered along with the two operations of addition $(+)$ and multiplication $(\cdot)$. We will spend some time trying to understand how $\mathbb{Z}$ is put together with respect to these two operations, and we will identify several key properties. We will then take a selection of those properties, the so-called ring axioms, and eventually aim at studying all structures that are defined by requiring a set $A$ along with two operations (which will be called $+$ and $\cdot$ even if they may have nothing to do with the conventional $+$ and $\cdot$) to satisfy the ring axioms. These structures will be called rings: from this perspective, $\mathbb{Z}$ is a particular example of a ring. Other examples being $\mathbb{Q}$ (rational numbers), $\mathbb{R}$ (real numbers), $\mathbb{C}$ (complex numbers); but many more exist, and most of them have nothing to do with numbers.
August 1, 2024 |
Section: Notes
Part I: Essentials Newton’s Laws of Motion Space and Time Newton’s three laws of motion are formulated in terms of four crucial underlying concepts: the notions of space, time, mass, and force. We begin by reviewing space and time.
Space Each point $P$ of the three-dimensional space we live in can be labeled by a position vector $\bf{r}$ which specifies the distance and direction of $P$ from a chosen origin $O$. The most natural way to identify a vector is in terms of its components, which are in the directions formed by our orthonormal basis. We can do this by introducing unit vectors for each of these coordinate axes.
August 1, 2023 |
Section: Notes
Introduction Part 1: Historical Introduction The Two Basic Concepts of Calculus The remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for tackling a variety of problems that arise in:
Physics Astronomy Engineering Chemistry Geology Biology Social Sciences Calculus is more than a technical tool, it is a collection of fascinating ideas that have interesting thinking human-beings for centuries. These ideas have to do with speed, area, volume, rate of growth, continuity, tangent line, and other myriad concepts. Another remarkable feature of the subject is its unifying power. Most of these ideas can be formulated so that they revolve around two rather specialized problems of a geometric nature. We proceed with a brief description of these problems.