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Introduction to Mathematics

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

G. H. Hardy, 1940

The definition for mathematics is not as clear-cut as one might think.

  • As the study of numbers - this is only partially true. Mathematics is not just about numbers, but also about shapes, patterns, and relationships. Many things deal with non-numerical concepts, such as geometry, topology, logic, and set theory.
  • As a deductive science - this is also only partially true. Mathematical reasoning is not just deductive, but also inductive, sometimes even abductive. Many things deal with inductive reasoning, such as statistics, probability, and combinatorics. Rather than being "pure" or "applied", mathematics is a spectrum of both, and various fields of mathematics can lie anywhere on this spectrum.
  • As the "science of formal systems" (Haskell Curry in 1951) - this does not capture the full essence of mathematics. Mathematics is not just about formal systems, but also about intuition, creativity, and discovery. Pure math without intuition is soulless.

However, the definitions aside, there are some common themes in mathematics:

  • Mathematics is about abstracting away the unnecessary details and focusing on the essential properties of objects. The first thing you learn, numbers, are abstractions of the real world. The number 1 is not a physical object; you can't say "give me 1". Instead, numbers, are abstractions of counting, and they have properties that are useful in various contexts.

    One advantage of abstraction is that it allows us to study a wide range of objects with similar properties. For instance, the same differential equations that describe the motion of a pendulum can also describe the flow of heat in a metal rod.

  • Mathematics is logical. It is about making precise arguments and drawing conclusions from them. You start with a set of axioms, which are assumed to be true, and then use logical reasoning to derive theorems. The axioms are the starting point, and the theorems are the conclusions. The process of deriving theorems from axioms is called proof. However, mathematics is not just about proving theorems; estimation, approximation, and conjecture are also important.

  • Mathematics is about patterns and structures. It is about recognizing patterns in the world and using them to build structures. For instance, the Fibonacci sequence is a pattern that appears in many natural phenomena, such as the branching of trees and the arrangement of leaves on a stem. The concept of a group is a structure that appears in many areas of mathematics, such as number theory, geometry, and algebra.

Generally, mathematics is a logical, abstract, and creative discipline that studies the world in a precise and structured way.

Discovery Process​

As previously mentioned, mathematics mixes together intuition and formulation. The intuition is the visual and intuitive interpretation of the concept, while the formulation is the more formal description that shows how the concept can be generalized and computed.

The general discovery process for mathematics is;

Specific Example → Intuition → Mathematical Model → Wide-Ranging Applications

This shows the advantage of abstraction. By abstracting things into their fundamental properties, we can apply the same concepts to a wide range of problems.

For instance, when Newton was studying the motion of the planets, he developed the concept of calculus to describe the motion of objects in general. Today, calculus is used everywhere, from physics to economics to computer science to just about every field you can think of.

One shortcoming of the modern education system is that it usually does this process in reverse. It starts with the formulation. This is probably partly responsible for the large distaste for mathematics. It's like learning to paint by starting with the chemical composition of paint.

Fields of Mathematics​

Mathematics is a vast field, with many different branches and subfields. First, we can divide mathematics into two main categories: pure and applied mathematics.

  • Pure mathematics is the study of mathematics for its own sake. It is about understanding the underlying principles and structures of mathematics. Pure mathematics is often more abstract and theoretical, and it is not necessarily concerned with practical applications.
  • *Applied mathematics is the study of mathematics for practical applications. It is about using mathematical concepts and techniques to solve real-world problems. Applied mathematics is often more concrete and practical, and it is concerned with finding solutions to specific problems.

To further classify mathematics, we can divide them into several branches. First, pure mathematics contains branches such as:

  • Logic—the fundamental building blocks of mathematics. It is about understanding the principles of reasoning and argumentation.

    First, in logic, we study the basic principles of logic. This includes basic logical operators and statement forms. For example, is a statement form that is true if both and are true. On the other hand, is not a statement form, but rather an expression that can be evaluated to a number.

    Then, we study the principles of logical reasoning. This includes the principles of deduction and induction. The principles of deduction are the principles of drawing conclusions from premises. For example, if is true and is true, then must also be true. On the other hand, induction is a probabilistic principle which suggests that if all observations of a phenomenon are true, then the next observation is also likely to follow the same pattern. Mathematics is wholly deductive, and it is important to understand the principles of deduction in order to understand the principles of mathematics.

    A notable result in logic is Gödel's incompleteness theorem, which states that any consistent formal system that is powerful enough to express arithmetic cannot be both complete and consistent. This means that there are true statements in the system that cannot be proven within the system. This theorem has profound implications for the foundations of mathematics and the limits of formal systems.

  • Set theory—the study of sets, which are collections of objects. It is about understanding the properties and relationships of sets. For example, is a set of numbers. The set is a set of numbers that satisfy the equation .

    We can combine sets in different ways. For example, the union of two sets and is the set of all elements that are in either or . The intersection of two sets and is the set of all elements that are in both and . The difference of two sets and is the set of all elements that are in but not in .

    Set theory also contains the study of infinite sets, which are sets that contain an infinite number of elements. For example, the set of all integers is an infinite set, as is the set of all real numbers. Infinite sets are also divided based on their cardinality, which is a measure of the "size" of the set. For example, the set of all integers has a cardinality of , while the set of all real numbers has a cardinality of . The study of infinite sets is a deep and complex topic, and it has many applications in various fields of mathematics.

  • Algebra—the study of mathematical symbols and the rules for manipulating them. The first topic within algebra one will encounter is linear algebra, which is the study of vectors and vector spaces. One will learn that vectors are elements of a vector space, which is a set of vectors obeying certain axioms. To manipulate vectors, we can add and scale them, as well as multiply them with the dot product and cross product. Furthermore, we can also define linear maps between vector spaces. The study of linear algebra is important in many fields, including physics, computer science, and economics.

    The next topic is abstract algebra, which is the study of algebraic structures such as groups, rings, and fields. These are much deeper structures that generalize the concepts of numbers and operations. For example, we learn that sets like , , and can be generalized into a field, which is a set of numbers with addition and multiplication operations that satisfy certain properties.

    Part of abstract algebra is representation theory, which is the study of how to represent algebraic structures in a more concrete way. For example, we can represent the group of rotations in 3D space as a set of matrices that represent the rotations.

  • Geometry—the study of shapes and their properties. The two main branches of geometry in pure mathematics are algebraic geometry and differential geometry.

    Algebraic geometry primarily focuses on the study of algebraic varieties, which are geometric objects defined by polynomial equations. For example, the set of all points in 3D space that satisfy the equation is a sphere.

    On the other hand, differential geometry is the study of smooth manifolds. A manifold is a topological space that is locally Euclidean, meaning that it looks like Euclidean space in small neighborhoods. For example, the surface of a sphere is a 2D manifold, while the surface of a torus is a 2D manifold with a hole in it. Physics suggests that the universe is a curved manifold, and hence we need differential geometry to study the universe.

  • Topology—one of the most abstract branches of mathematics. When two objects can be transformed into each other without cutting or gluing, they are said to be topologically equivalent. For example, a coffee cup and a donut are topologically equivalent, as they can be transformed into each other by stretching and bending. On a deeper level, we find that different spaces are not just topologically equivalent, but also homeomorphic, which means that they can be transformed into each other by continuous functions.

    Topological spaces are the most basic objects in topology. They are sets of points with a topology, which is a collection of open sets that satisfy certain properties. For example, the set of all points in 3D space is a topological space with the standard topology, which is the collection of all open sets in 3D space. We can introduce certain structures to topological spaces, such as the metric, which is a function that measures the distance between points in the space. Spaces with a metric are called metric spaces. After doing this more, we realize that the sets that we learn from other fields are fundamentally topological spaces with certain structures. For example,

Resources​

The main resources I used to create these notes are:

These are all excellent resources for learning (esp. self-learning) mathematics.